Advnced Anlyticl Model for the Prognostic of Industril Systes Subject to Ftigue Abdo Abou Joudé To cite this version: Abdo Abou Joudé. Advnced Anlyticl Model for the Prognostic of Industril Systes Subject to Ftigue. Mechnicl engineering [physics.clss-ph]. Aix-Mrseille Université; Lebnese University - EDST,. English. <tel-87464> HAL Id: tel-87464 https://tel.rchives-ouvertes.fr/tel-87464 Subitted on 8 Oct 3 HAL is ulti-disciplinry open ccess rchive for the deposit nd disseintion of scientific reserch docuents, whether they re published or not. The docuents y coe fro teching nd reserch institutions in Frnce or brod, or fro public or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de docuents scientifiques de niveu recherche, publiés ou non, énnt des étblisseents d enseigneent et de recherche frnçis ou étrngers, des lbortoires publics ou privés.
Université Libnise Université Libnise École Doctorle Sciences et Technologie LSIS O-ADVISED THESIS Between the LEBAESE UIVERSITY And AIX-MARSEILLE UIVERSITY École Doctorle en Mthétiques et Infortique de Mrseille - ED 84 To obtin the Ph.D. diplo in Autotic ontrol Advnced Anlyticl Model for the Prognostic of Industril Systes Subject to Ftigue Presented nd defended by: Abdo ABOU JAOUDÉ The Jury is coposed of: Pr. Foud Kddh Université Sint Joseph, Libn Rpporteur Pr. Didier Theilliol Université de Lorrine, Frnce Rpporteur Pr. Ahed El Hjjji Université de Picrdie Jules Verne, Frnce Exinteur Dr. lovis Frncis Université Libnise, Libn Exinteur Dr. Seifedine Kdry Université Libnise, Libn Directeur de thèse Pr. Hssn our UAE University, UAE Directeur de thèse Pr. Mustph Ouldsine Aix-Mrseille Université, Frnce Directeur de thèse Dr. Khled El-Twil Université Libnise, Libn Invité Thesis prepred t Lbortoire des Sciences de l'infortion et des Systèes (LSIS - UMR RS 796) - Frnce nd the École Doctorle Sciences et Technologie (EDST) - Libn
"The Ancient of Dys", Willi Blke, 794 i
ii Willi Blke
DES PESÉES ISPIRATES "e qu'il y d'incopréhensible, c'est que l'univers soit copréhensible." Albert Einstein "Le hsrd est le pseudonye de Dieu qund Il ne veut ps signer." Antole Frnce Si j i ppris une chose u cours de longue vie, c est que toute notre science, confrontée à l rélité, pprît priitive et enfntine et pourtnt, c est ce que nous possédons de plus précieux. Albert Einstein S scrée jesté le Hsrd décide de tout. Voltire Toute pensée éet un coup de dés. Stéphne Mllré Le théticien, eporté pr son cournt de syboles tritnt de vérités pureent forelles, peut cependnt obtenir des résultts d une iportnce infinie pour notre description de l univers physique. Krl Person Ainsi, joignnt l rigueur des déonstrtions de l science à l incertitude du sort, et concilint ces deux choses en pprence contrdictoires, elle peut, tirnt son no des deux, s rroger à bon droit ce titre stupéfint: l géoétrie du hsrd. Blise Pscl iii
"Dieu est subtil, is il n'est ps lveillnt." Albert Einstein "Une intelligence qui, à un instnt donné, connîtrit toutes les forces dont l nture est niée et l sitution respective des êtres qui l copose ebrsserit dns l êe forule les ouveents des plus grnds corps de l'univers et ceux du plus léger toe ; rien ne serit incertin pour elle, et l'venir, coe le pssé, serit présent à ses yeux." Mrquis Pierre-Sion de Lplce "Le plus beu sentient du onde, c est le sens du ystère. elui qui n jis connu cette éotion, ses yeux sont ferés." Albert Einstein iv
TABLE OF OTETS Generl Introduction... I Introduction to Systes Prognostic... 5 I. Introduction... 6 I.. Mintennce Evolution... 6 I.. Mintennce Optiiztion... 8 I. Intelligent Mintennce... 9 I.3 Degrdtion Prognostic... I.3. Degrdtion versus Prognostic... I.3. Equipent Degrdtion Trjectory... I.3.3 Definition nd Methodologies... 5 I.4 Prognostic Definition... 7 I.5 The Role of Prognostic in Lifetie Process... 8 I.6 Stte-of-the-Art of the Prognostic Approches... 9 I.6. Prognostic Bsed on Models... I.6.. Advntges nd Drwbcks of the First Approch... 5 I.6. Prognostic Guided by Dt... 6 I.6.. Prognostic by Trend Anlysis... 7 I.6.. Prognostic by Lerning... 8 I.6..3 Prognostic by Stte Estition... 3 I.6..4 Advntges nd Drwbcks of the Second Approch... 3 I.6.3 Prognostic Bsed on Experience... 3 I.6.3. Stochstic Approch... 33 I.6.3. Relibility Approch... 34 I.6.3.3 Advntges nd Drwbcks of the Third Approch... 37 I.6.4 Methodology Bsed on Abci of Degrdtion... 38 I.7 Sury... 4 I.8 onclusion... 43 References... 44 II Anlytic Liner Prognostic Model of Dynic Systes... 53 II. Introduction... 54 v
II. Proposed Prognostic Model... 55 II.. Dge Evolution Lw... 56 II.. Pris-Erdogn's Lw... 57 II..3 Plgren-Miner's Rule... 59 II..4 WÖhler's urve... 59 II..5 Stress Intensity Fctor... 6 II..6 Additivity Rule in Plgren-Miner's Rule... 6 II..7 Mintennce nd Dignostic/Prognostic... 63 II..7. Flowchrt of Vrious oponents of Dignostic/Prognostic/Mintennce Process... 64 II..7. ycle of Prognostic-Dignostic-Mintennce... 65 II..8 Accuultion of Ftigue Dge... 66 II..9 Flowchrt of the Prognostic Model... 69 II.. Environent Effects in the Proposed Prognostic Model... 7 II.3 Appliction of the Prognostic Method to Industril Systes... 7 II.3. Vehicle Suspension Ftigue Life... 7 II.3.. Types of Mechnicl Effects, Their Mechniss, nd Possible onsequences... 76 II.3.. Autotic Dignostic of Bd Suspension Bushing... 76 II.3..3 Prognostic Study for Vehicle Suspension Systes... 77 II.3..4 Syste Identifiction... 78 II.3..5 Ftigue Dge Modeling of Suspension... 79 II.3..6 Siultion of the Degrdtion Model... 8 II.3..7 Siultion of Three Rod Profiles... 8 II.3..8 Siultion Results... 83 II.3..9 Anlysis of the Siultion Results... 86 II.3.. onversion of RUL into Yers... 87 II.3. Prognostic Study for Pipelines Systes... 88 II.3.. Introduction... 88 II.3.. Pipes Stress Modeling... 89 II.3..3 Stte of Stresses in the Tube Body... 9 II.3..4 Stress Intensity Fctor... 9 II.3..5 Degrdtion Model Expression of Pipes... 9 II.3..6 Siultions of Three Levels of Internl Pressure... 9 II.3..7 Unburied Pipe se..... 93 vi
II.3..8 Buried Pipe se... 95 II.3..9 Offshore Pipe se...98 II.4 onclusion... References... III Anlytic onliner Prognostic Model of Dynic Systes... 5 III. Introduction... 6 III. Stte-of-the-Art: onliner Dge Accuultion... 7 III.. Dge Theories Bsed on Endurnce Liit Reduction... III.3 onliner-dge-bsed Prognostic... III.3. Disdvntges of Liner Dge Accuultion... 3 III.3. Double Liner Dge Rule (DLDR)... 3 III.3.3 Dge urve Approch (DA)... 4 III.3.4 Double Dge urve Approch (DDA)... 5 III.4 onliner uultive Dge Model... 6 III.4. Solution of the Differentil Eqution of Degrdtion... 7 III.4. Reltion between D nd t Specific ycle... 9 III.4.3 Recursive Reltion of onliner Dge D... III.5 Appliction to Suspension Syste... III.5. Results of the Siultion... III.5. onversion of RUL into Yers... 4 III.5.3 oprison with the Liner se... 5 III.5.4 Advntges of the Proposed Model... 6 III.6 Appliction to Pipeline Syste... 7 III.6. Unburied Pipe se... 8 III.6.. oprison with the Liner se... 3 III.6. Buried Pipe se... 3 III.6.. oprison with the Liner se... 34 III.6.3 Offshore Pipe se... 34 III.6.3. oprison with the Liner se... 37 III.6.4 Vlidtion of the Pipelines Lifeties... 38 III.7 onclusion... 39 References... 4 vii
IV Stochstic Liner nd onliner Anlytic Prognostic Model... 43 viii IV. Introduction... 44 IV. Stte-of-the-Art: Stochstic Ftigue Modeling... 44 IV.. Definition of the J-Integrl... 48 IV.3 Stochstic Liner Dge Accuultion... 5 IV.4 Stochstic Modeling... 5 IV.5 Stochstic RUL... 5 IV.6 Relibility Evlution of Dge Stte... 53 IV.7 Stochstic Bsic Preters... 55 IV.7. Initil rck Width... 55 IV.7. PDF of rck Length t Loding ycle... 56 IV.7.3 PDF of the Initil Dge D... 59 IV.8 Eqution of the Stochstic-Bsed Prognostic... 6 IV.8. Developent of d D ~... 6 IV.8. Developent of d ~... 6 IV.9 Flowchrt of the Stochstic-Bsed Liner Prognostic... 6 IV. Appliction to the Suspension Syste... 63 IV.. Liner Stochstic se... 64 IV... One Rndo Vrible... 64 IV... onversion of Lifeties into Yers... 65 IV... Two Rndo Vribles... 66 IV... onversion of Lifeties into Yers... 68 IV... oprison: Deterinistic - Stochstic Results (for Liner Dge Lw)... 68 IV...3 RUL Evlution of Suspension in Stochstic se... 69 IV...3 Vlidtion of the Suspension Life under Liner Dge Rule... 7 IV.. onliner Stochstic se... 7 IV... Stochstic onliner uultive Dge... 7 IV... Flowchrt of the Stochstic-Bsed onliner Prognostic.. 7 IV...3 One Rndo Vrible... 73 IV...3. onversion of Lifeties into Yers... 73 IV...4 Two Rndo Vribles... 74
IV...4. onversion of Lifeties into Yers... 76 IV...4. oprison: Deterinistic - Stochstic Results (onliner Dge Lw)... 76 IV...5 Vlidtion of the Suspension Life under onliner Dge Rule... 78 IV. Appliction to the Pipeline Systes to Three ses... 78 IV.. Eqution of the Stochstic-Bsed Prognostic... 78 IV.. Genertion of Internl Pressure P i... 79 IV... Monte-rlo Siultion Principle... 8 IV... Model A: Unifor Genertion of Tie t... 8 IV...3 Model B: One Initil Tringulr Period T P... 8 IV...4 Model : Multi-Tringulr Period... 8 IV..3 Liner se of Dge... 83 IV..3. One Rndo Vrible (Pressure)... 83 IV..3.. Model A for Pressure Genertion... 83 IV..3.. Model B for Pressure Genertion... 85 IV..3..3 Model for Pressure Genertion... 87 IV..3. Two Rndo Vribles: Pressure (One Tringulr Period)- (Lognorl Lw)... 88 IV..4 onliner se... 9 IV..4. One Rndo Vrible (Pressure)... 9 IV..4. Two Rndo Vribles (Pressure nd Initil rck Length)... 94 IV..4.. oprison: Deterinistic - Stochstic Results (onliner Dge Lw)... 98 IV..5 Vlidtion of the Pipelines Lifeties in Stochstic onditions... 99 IV. onclusion... References... onclusion nd Future Works... 5 List of Publictions... 9 Thesis Abstrcts... Résué de l Thèse... 3 ix
GEERAL ITRODUTIO Due to technologicl dvnces nd to incresing copetitiveness of countries of low production costs, the industril sectors of developed countries hve to fce constntly new chllenges which re incresingly difficult. These chllenges hve s principl objective the xiiztion of copetitiveness by the reduction of production costs, the ugenttion of the instlltions profitbility, nd the cretion of innovtive products by gurnteeing stff nd equipents security, nd by respecting the regultions in ters of environentl requireents. The developent of solutions cpble of iproving the production systes perfornces is then necessry in order to intin the production sites survivl t the hert of the developed countries []. Industry is one of the engines of the econoic developent of country. The perfornce ws lwys jor preoccuption of copnies. owdys, its evlution is not only function of productivity but lso of flexibility, costs, delys, qulity, sfety, socil perfornces, environentl perfornces, etc. We hve shifted then fro one-criterion-evlution to ulti-criteri-evlution tht cn extend the products coplete life cycle. We spek then of globl perfornces nd long-lsting developent. Mintennce is thus strtegic point in the copetitiveness progress nd iproveent. Hence, intennce knows nowdys spectculr upswing. In fct, intennce provides the possibility of exploiting enterprise resources in order to iprove their perfornces by optiizing the utiliztion of hun nd teril ens. Since its beginning, intennce hs not cesed to progress nd iprove due to the eergence of Infortion nd ouniction Technologies (IT) s well s due to the requireent nd exigency iposed by the worldwide econoic context. Mintennce hs becoe true discipline with its own ethodologies nd concepts. To ke the clssicl strtegies of intennce ore efficient nd to tke into ccount the evolving product stte nd environent, prognostic odels need to be developed s copleent of existent intennce strtegies. When the intennce strtegy includes prognostic function of the equipent reining useful lifetie, we spek of Prognostics nd Helth Mngeent (PHM), doin fro which hs eerged the "PHM society".
The prognostic is quite new re of interest, it is the bility to predict nd prevent possible fult or syste degrdtion before filures occur. Actully, If it is possible to predict the condition of chines nd systes, intennce ctions cn be tken hed of tie. As result, iniu downtie cn be chieved. Prognosis hs been defined s prediction of when filure y occur i.e. ens to clculte the Reining Useful Lifetie (RUL) of n sset. In order to ke good nd relible prognosis it ust hve good nd relible dignosis. As recent discipline, prognostic is key sub-process for the proctive intennce [] for Mintining systes in Opertionl ondition (MO). The integrtion of prognostic function in proctive intennce process llows in dvnce, gurnteeing to respond to the different tsks ssigned to the syste, nd to prevent functioning brekdown s well s expensive intennce interventions. Let us tke for exple ship king journeys for severl weeks; it is ore pproprite to chnge n equipent or to ebrk good replceent equipents before strting the journey thn to ke intennce intervention on the other side of the plnet [3]. The systes jor prt (plnes, ships, vehicles, petrocheicl systes, etc.) presents big coplexity in ters of their hybrid chrcter. The continuous spect of the echnicl prts (degrded filure: ftigue for instnce) is lrgely relted to the discrete spect of the electric nd electronic prts (binry filure: On/Off). They re systes tht contin lrge nuber of vribles hving coplex reltionships; hence, they re clled: coplex systes. Wheres there exists nowdys for the doin of dignostic instruents tht integrte the notion of systes due to experience nd ethods cquired in the lst decdes. Few tools or very specific tools re vilble in the prognostic doin. Most of the publictions on this topic present prognostic in the frework of n eleentry syste. The objective of dignostic is to detect nd to explin the occurrence of syste filure or brekdown wheres the objective of prognostic is to predict the future stte of degrdtion of syste extrpolted fro its current stte. In the cse of dignostic we wlk bckwrd in tie, wheres in the cse of prognostic we wlk forwrd in tie, or in other words, we nticipte tie.
Moreover, predicting the reining useful lifetie of industril systes becoes n iportnt i for industrilists to overcoe the occurrence of sudden filures tht cn led to very expensive consequences. Then, the recent prognostic pproches try to copenste for the inconveniences enting fro clssicl intennce strtegies becuse they neglect the evolving product stte nd environent. The erlier recent developents in syste design technology like in erospce, defense, petro-cheicl nd utootive industry hve the gol to ensure their high vilbility. In the Autotic ening of the ter, prognostic is generlly ssocited with the notion of degrdtion which represents the ccuultion of the syste wer out. A prognostic consists of predicting the future evolution of degrdtion by tking into considertion the fctors tht odify the degrdtion dynics. These fctors cn be subdivided into two ctegories: the fctors linked to the solicittion of the syste (rod excittion in, gs pressure in MP, etc.) nd those linked to the environent in which the syste evolves (huidity, teperture, soil pressure, etc.). Usully, the influence of these two coponents on degrdtion is not very well known or even totlly ignored. Vrious ethods hve been pplied to the prognostic of degrded coponents. Generlly, they re clssified in three fundentl filies: - The pproches bsed on odels (Model-bsed prognostics) - The pproches guided by dt (Evolutionry or trending odels) - The pproches bsed on probbilistic techniques (Experience-bsed prognostics) The odel-bsed prognostic pproch is very precise becuse it hs inly two dvntges: the cpcity of including the systes physicl infortion nd the cpcity of redpttion to ny new infortion. The dt-driven pproch requires lrge nd relible dt sple in order to be ccurte. The experience-bsed pproch is well dpted to coplex systes but requires n excellent historic dt, lrge feedbck nd expert knowledge. The new prognostic procedure proposed in this work belongs to the first pproch. This thesis is dedicted to the prognostic evlution of dynic systes. The work presented here is t developing n dvnced tool to tret the prognostic evlution in liner 3
nd nonliner deterinistic context in first prt s well s in the stochstic context in second prt. Our purpose is to prepre generl prognostic tool tht cn be cpble of well predicting the RUL of syste bsed on n nlyticl dge ccuultion lw in either deterinistic or stochstic context. hpter I is devoted to generl prognostic stte-of-the-rt tht encopsses the prognostic pproches existing in specilized literture. hpter II defines the dopted dge criterion nd dge ccuultion then develops recursive odel expressed in ters of degrdtion index bsed on liner spect of dge ccuultion. In order to illustrte the presented ethodology, the siultion of n utootive suspension syste is considered. Then, siultion of petrocheicl pipelines is illustrted in three odes: unburied, buried, nd offshore. hpter III introduces nonliner odel for dge ccuultion followed by the se pplictions. Finlly, hpter IV expnds the proposed deterinistic prdig to stochstic doin. The two pplictions to suspensions nd pipelines re considered in this finl chpter. References [] K.M. GOH, B. TJAHJOO, T.S. BAIES, nd S. SUBRAHMAIA, "A Review of Reserch in Mnufcturing Prognostics", In 6 IEEE Interntionl onference on Industril Infortics, pp. 4-4, ew York, USA, August 6. [] A. MULLER: ontribution à l intennce prévisionnelle des systèes de production pr l forlistion d'un processus de pronostic. Thèse de doctort, Université Henri Poincré - ncy I, Frnce, Juin 5. [3] F. PEYSSO, ontribution u pronostic des systèes coplexes, thèse de doctort, Université d Aix-Mrseille, Frnce, Décebre 9. 4
HAPTER I ITRODUTIO TO SYSTEMS PROGOSTI 5
I. - Introduction In the current chpter we present the evolution of intennce in order to introduce the concept of intelligent intennce nd the role of Prognostics nd Helth Mngeent during the syste life cycle. It develops lso the stte of the rt of prognostic pproches: odel-bsed prognostic, dt-bsed prognostic, nd experience-bsed prognostic. This stte of the rt pves the wy for the present work nd contribution to this field. Whether in the doin of echnics or in civil engineering or in electronics, the desire nd the need to ke dignostic s precise s it cn be nd to cquire rel cpcities of prognostic, exist since the first hun exploittion of expensive nd coplex chines. This otivtion led to gret nuber of scientific nd industril works in the purpose to develop nd ipleent different levels of dignostic nd prognostic nd hence to optiize intennce strtegies []. Mintennce ctivities hve lwys existed. At the beginning, they consisted of n intervention fter syste filure. But rpidly, the unpredicted nd soeties very long shutdowns, due to intennce interventions, were found to be very expensive. Therefore ore dvnced intennce strtegies hve evolved nd were fterwrd developed. I.. - Mintennce Evolution The different intennce concepts cn be clssified into three big ctegories which re: corrective intennce, preventive intennce, nd predictive intennce. The corrective intennce is the intennce tht intervenes fter the occurrence of filure in the syste, wheres the preventive intennce is relized when the syste is currently functioning []. It is iportnt to note tht corrective opertions intervene only when filure occurs, wheres preventive intennce cn be progred in function of different preters. Predictive Mintennce (PdM) techniques help deterine the condition of in-service equipent in order to predict when intennce should be perfored. This pproch offers cost svings over routine or tie-bsed preventive intennce, becuse tsks re perfored only when wrrnted. The in vlue of Predicted Mintennce is to llow convenient scheduling of corrective intennce, nd to prevent unexpected equipent filures. The key is "the right infortion in the right tie". By knowing which equipent needs intennce, intennce work cn be better plnned (spre prts, people etc.) nd wht would hve been "unplnned stops" re trnsfored to shorter nd fewer "plnned stops", thus incresing plnt 6
vilbility. Other dvntges include incresed equipent lifetie, incresed plnt sfety, fewer ccidents with negtive ipct on environent, nd optiized spre prts hndling. The concept of corrective intennce hs the gol of resetting the syste to its norl functioning stte fter the occurrence of its filure. During the seventies, the concept of preventive intennce hs ppered, nd it hs the gol of reducing the probbility of filure s well s to optiize the costs relted to the syste usge. One of the first used strtegies ws the systetic intennce tht consists of executing regulr interventions t equl tie intervls, following n priori nd well deterined schedule. The optiiztion of such strtegy consists of evluting the opertions periods lbeit in preventing the syste filure by following very frequent opertions. The syste vilbility is thus incresed but finncilly this strtegy reins not very rewrding nd ny studies hve shown tht the usge tie is not the only fctor leding to filure occurrence. The periodicity of interventions cn be clculted in function of tie or of the nuber of usge units (nuber of functioning cycles, nuber of kiloeters, nuber of nufctured products, etc...). Since the eighties, due to the evolution of infortion resources, new intennce strtegies were born. Their principle consists of using rel-tie infortion in order to onitor continuously certin significnt preters of degrdtion or of syste perfornce. We spek then of conditionl intennce. The interventions plnning rely then on the existence nd deterintion of the criticl thresholds of these significnt preters; hence, we spek of decision thresholds. Thus, the predictive intennce ppers. It is subordinted to the nlysis of the surveyed evolution of the significnt preters of degrdtion. The estition of the output of this preters onitoring, llows to dely or to speed up intennce interventions. The conditionl nd predictive intennces ssue tht the intervention will occur before the occurrence of the filure of the onitored syste evolution. This is why, during the nineties, new ethodologies, clled proctive intennce, were invented in order to onitor continuously not the syste evolution but the evolution of priry cuses of filure occurrences of the onitored syste. 7
It is iportnt to note tht during the period of the evolution of intennce strtegies, we observe lso chnge in intennce ngeent. In fct, distnt intennce hs rpidly evolved nd dvnced locl intennce due to couniction networks. Following the Internet big bng, the concept of distnt intennce hs trnsfored to e- intennce [3]: it is concept tht uses web services for better coopertion ong the different coponents of intennce, for better shring of knowledge, nd follow up in rel tie of the syste fro nywhere round the world. The eergence of these concepts nd the econoic context llowed the enterprises to externlize this service by using specilized gents. I.. - Mintennce Optiiztion The intennce optiiztion consists of finding iddle point between preventive intennce nd corrective intennce, ll this by respecting fixed objectives. The intennce interventions dtes re then deterined in wy to optiize criterion relint on intennce cost, on equipents vilbility, s well s on security, or ore on coproise ong the three of the. Moreover, if we hve ny wys of onitoring ny finncil resources, nd if we replce very frequently the syste equipents, then we will observe few filures. On the contrry, if we dispose few finncil ens, nd we don't do the equipent intennce, then we will observe gret nuber of filures. It sees evident tht the filure costs re inversely proportionl to the intennce costs. In fct, the oney sved due to less intennce will be spent on the interventions for the syste recovery in order to return to its norl stte. The bsence of syste intennce leds eqully to syste filures in chin. The su of the costs of intennce nd filures represents the totl cost to intin the syste functioning. An optil intennce is intennce tht iniizes t the se tie the costs relted to systetic intennce nd the costs relted to syste recovery fter filure. This optil intennce cn be ttined by using helping utoted syste to intennce in order to identify the equipents tht hve to be intined nd sustined. This first nlysis shows tht there exists n incresing interest in intelligent intennce in which surveillnce occupies fundentl plce [4]. In the scientific counity, principlly in the Autotic nd Artificil Intelligence counities, surveillnce led nd is still leding to big nuber of reserch nd works. These works hve eqully evolved with tie, strting fro siple detection of bd functioning, pssing by filures 8
dignostic nd degrdtion dignostic, nd is oriented nowdys to prognostic nd the prediction of degrdtion nd filures. The following section presents the intelligent intennce s well s the principl concepts nd the notion of degrdtion for prognostic is then introduced, followed fterwrd by the stte of the rt of the known pproches to prognostic. At the end of this chpter, sury of the different pproches is presented. I. - Intelligent Mintennce As we hve lredy discussed in the previous prgrph, the intennce function cnnot be reduced to the sole ctivity of intennce of set of chines. It hs lso the tsk to intervene during the whole syste exploittion cycle: the choice nd the conception of the teril, the deterintion of the intennce plns, the orgniztion nd the logistic of the intennce ctivities, the follow up nd the nlysis of the syste evolution, the prediction of the syste future evolution, etc. The intelligent intennce differs fro the trditionl policies of intennce which re bsed on sttic threshold of lr. The power of intelligent intennce lies in the nlysis nd the follow up of the helth of the equipents coing fro set of dt inferred fro the ERP (Enterprise Resource Plnning), the MPA (Mngeent of Production Assisted by oputers), the MMA (Mngeent of Mintennce Assisted by oputers), or even fro surveillnce systes which re bsed on the esureents of physicl vribles provided by sensors. This dynic follow up of the perfornces nd of the syste stte of degrdtion requires the cquisition, the centrlized ngeent, the vlidtion, nd finlly the nlysis of the huge set of dt of very different nture. Appering t the beginning of the third illenniu, the ter Prognostic nd Helth Mngeent (PHM) ws defined s n pproch tht uses esureents, odels nd lgoriths to detect filures, to evlute the helth nd to predict the syste degrdtion evolution [5]. The PHM is sustining pproch during the whole syste life cycle, nd whose objective is to reduce, even lso to eliinte the inspections of the syste nd the intennce t regulr intervls, by using onitoring nd prediction instruents dedicted nd relted to the logistic chin of the syste, leding hence to n unprecedented rectivity. Inheriting the principles of ondition Bsed Mintennce (BM), the concept of PHM expnds its cpcities nd proposes robust frework for the optiiztion of intennce nd of the logistic in order to increse the opertionl vilbility of the syste. 9
A odern tool of PHM cn include gret nuber of functions [6] such s: - The detection nd the isoltion of filures - Advnced lgoriths of dignostic nd prognostic - Algoriths of filures nd degrdtion tolernce - Estition of the reining useful lifetie of n equipent - The follow up of the helth nd/or of the degrdtion of n equipent - The filtering: the lrs nd infortion ngeent by yielding the right infortion to the right person t the right tie - Helping lgoriths to the decision king for the syste ngeent - Etc. The jor prt of these functions is the evolutions of the functions put in order in onitoring nd dignostic systes [7]. Bsed on the concepts of the ngeent of equipents helth, the tool of PHM uses these functions in copleentry wy in order tht they hve better ipct on intennce ctivity, rther thn by using the ech one lone. Even if ech of these functions is developed nd iproved t the se tie s the tool of PHM, the prognostic represents new function which sees to be very difficult nd even to be risky fro technologicl point of view [8]. In literture, the PHM pproch of intennce is usully represented by the cycle PHM [8,9]. One of the in differences is the positioning of the dignostic reltively to prognostic. The ipleenttion of the PHM pproch is done in two phses: - A first phse tht hs the objective of studying which fctors ct on the syste helth nd how they influence it. This study llows deterining which helth indictors pertin for the considered syste nd to estblish the dequte dignostic nd prognostic lgoriths. - A second phse tht consists in the integrtion nd the ipleenttion of the tools deterined in the first phse. The first step is step of signl processing in order to extrct the syste helth indictors. These indictors re used by the step of onitoring to estite the syste current helth stte. The current helth stte serves then s strting point for the dignostic nd prognostic lgoriths whose corresponding gols re the isoltion of filures nd the prediction of the syste helth evolution. The outputs of these lgoriths re used to djust the pln of intennce nd/or to odify the syste control preters. We spek then of tolernce to filures or to degrdtion.
The Prognostic is currently one of the ost difficult spects in the PHM cycle s well s the spect hving the biggest potentil in ters of reducing the costs of functioning nd of logistic during the whole lifetie cycle of coplex syste, even in ters of iproving its vilbility nd security []. With the dvent of the prognostic techniques, we observe eqully chnge in the behvior of the industrilists who do not buy nyore nowdys intennce service but who buy n vilbility chine. I.3 - Degrdtion Prognostic I.3. - Degrdtion versus Prognostic Dignostic nd prognostic re two words of Greek origins. In the etyologicl sense, dignostic is the cquisition of knowledge fro observble signs, wheres prognostic is precognition or knowing in dvnce. In the utotic sense, the ening of the two words is ore precise nd technicl. Dignostic consists in the regression in tie in order to explin why the syste is in given stte t instnt t. Prognostic consists of nticipting in tie in order to predict the syste future stte t the instnt t + t. Dignostic nd prognostic re two prllel processes tht cn be used copleentrily or seprtely []. Dignostic nd prognostic rein intrinsiclly linked by the chin "cusesconsequences", s well s by the concepts tht they nipulte to lern: defects, filures nd degrdtions. These lst concepts hve in literture different definitions for different uthors. We will use the following definitions []: - A fult is the devition in behvior between n observed chrcteristic nd theoreticl chrcteristic. - A filure is the inbility of n equipent to ccoplish its function. - A degrdtion is the wer out of the equipent, nd the decrese of its perfornces. Fult nd filure re concepts tht we qulify s discrete since they represent stte of the equipent, wheres degrdtion is continuous concept tht evolves during the equipent lifetie.
I.3. - Equipent Degrdtion Trjectory The Assurnce In Functioning (AIF) is nowdys discipline lrgely used in order to predict equipent filures. The coponent of this discipline dedicted to prediction is the relibility tht chrcterizes the probbility tht n pprtus ccoplishes required function in given conditions, nd during given tie []. The grphs of oriented sttes re tool used by relibility experts in order to represent the evolution of equipents sttes. The nodes of the grph constitute the equipent sttes nd the rcs represent the trnsition ong sttes (figure.). On-Line Monitoring (OLM) Dt Monitoring nd Detection Is there n noly or fult? Dignostics Wht is the fult or degrdtion? Prognostics Wht is the Reining Useful Life (RUL)? Risk Mitigtion How cn the effects of degrdtion be itigted? Dignostics nd Prognostics Figure. - Dignostic-Prognostic hin of "uses-onsequences". The stte "ew" represents the equipent newly coing fro the fctory. It is phse whose objective is to eliinte the initil fults. Following this phse, the equipent is put in service nd is integrted in set in order to function in its noinl stte. When the equipent reches the end of its life, it psses to the stte of fult. In the fult stte, the equipent is still functioning but in non-noinl wy nd with reduced perfornces, till it psses to the stte of filure where it is no ore functioning. When the equipent is in the stte of fult or the stte of filure, n opertion of intennce llows restoring the equipent noinl stte (figure.).
The relibility counity hs discrete vision of the equipent life to the contrry of the utotic counity of PHM tht chrcterizes the life of n equipent by continuous vrible. The ebers of the utotic counity consider tht degrdtion is process tht evolves during the whole equipent lifetie till it ttins criticl threshold of fult tht leds to the stte of filure. This vrible is generlly n indictor of helth or of degrdtion of the equipent nd tht is norlized between nd where degrdtion is the copleent of the of helth. Put in service Fult Fult ew oinl Fult Filure Mintennce Mintennce Progression of Degrdtion Figure. - Oriented Grph of the Equipent Life Sttes A degrdtion trjectory is defined in stte spce s the wy followed by the degrdtion stte, in function of the odes of equipent degrdtion. Most of the equipents hve ny odes of degrdtion, where ech ode hs unique trjectory [5]. The objective of PHM tools is to follow nd to updte the rel degrdtion trjectory of given equipent nd to predict the evolution of this trjectory in function of the future usge of the equipent [] (figure.3). In dge theory, there exist two types of degrdtion: isotropic nd non-isotropic. The odels of isotropic degrdtion re the siplest odels of dge theory, where the nonliner degrdtion behvior is represented by one internl vrible [3]. This vrible cn be considered s degrdtion indictor. In the cse of non-isotropic degrdtion odels, the nonliner degrdtion behvior is represented by tensor [4]. In the PHM pproches, we consider usully the isotropic odels, becuse they re generlly sufficient in order to chieve 3
good prediction nd esureent of the reining useful lifetie of n equipent [5]. Ech scientific discipline hs its own proper odels, but whtever the concerned phenoenon, the degrdtion trjectory enting fro these odels, dopts either liner, concve, or convex for (figure.4). Degrdtion esureent Degrdtion threshold Estited degrdtion lw (fro the helth stte) Anlysis of helth stte (bnorl solicittions: fults,...) oinl degrdtion lw Tie t Estited RUL(t) RUL(t) Syste Lifetie Degrdtion lw for prognostic Figure.3 - Estited nd oinl Degrdtion Trjectory of n Equipent. Filure level Degrdtion index onvex Liner oncve Lifetie Figure.4 - Different Trends for Degrdtion Trjectory. 4
I.3.3 - Definition nd Methodologies In literture, fro one uthor to nother, the definition of prognostic chnges [6-], but they ll gree on one point: prognostic is process encopssing cpcity of prediction. The in difference ong the proposed definitions is the horizon on which this prediction is perfored. For soe uthors, prognostic is the cpcity to detect nd isolte the newborn defects or even the eleents leding to defects. For others, prognostic is the cpcity to estite the reining useful lifetie (RUL) of n equipent in function of its functioning history nd its future usge. The reining lifetie is typiclly defined in ters of tie, of chrge cycle, or of ission [5]. In the first cse, the horizon of prediction is the short ter since the defect lredy exists, wheres in the second cse, the horizon is the long ter. The expression "predictive dignostic" is ore explicit in the first cse [7]. Whtever the ethodology used for prognostic or predictive dignostic, the notion of degrdtion is n intrinsic eleent since it chrcterizes the equipent usge. The predictive dignostic cn be considered s being the dignostic of degrdtion stte, where the degrdtion stte is sub-stte of the equipent noinl stte. A notion eqully linked to prognostic is the notion of uncertinty since it is very difficult to predict the future in sure nd certin wy []. The nlysis of different ethodologies of prognostic in literture llows us to put in evidence two principles of prognostic pproches. The difference between the two principles is situted t the level of usge of the degrdtion vribles in direct or indirect wy. In the first principle of pproch, where these vribles re used, the process of prognostic is bsed on the concept of degrdtion trjectory. It consists of estiting the evolution of the trjectory fro the vilble given dt nd to ke this trjectory evolve in the future by using or not the future utiliztion conditions of the equipent. In the second principle of pproch, we do not seek to know the level of the equipent degrdtion. It consists of estiting, then to extrpolte n indictor, such s for exple the RUL, fro the observtions of the equipent output vribles. The behvior of the equipent is represented by n input vrible vector U, n output vrible vector Y, nd three functions tht express: 5
- The evolution of the internl vribles tht chrcterizes the equipent dynics, its behvior in function of the input vribles, in function of its environent, nd in function of its degrdtion stte. - The evolution of the degrdtion vribles. It is this evolution tht defines the degrdtion trjectory of the equipent. It is conditioned by the usge of the equipent nd chrcterized by its environent nd its input vribles s well s by the internl vribles. - The output function tht defines the output vribles fro the internl vribles. The output vribles re directly observble on the contrry to the internl vribles. Process: Sense Dignose Prognose Decide Deterine: Vibrtion Trends Liits for relible Short-ter Teperture Devitions perfornce. rediness (dys) Pressure Fult loction Probbility of filure - ission pln Fuel flow Fult clssifiction Reining useful life Long-ter Speed Dge echnis rediness (yrs.) Mteril dge - wer (high probbility) - intin/inspect - crcking (low probbility) - retire/replce Tools: Acceleroeters Pttern recognition Expert systes Risk nlysis Therocouples eurl nets - experience Decision nlysis Pres. trnsducers On-line-DE (on Destructive Evlution) Hun senses Physics odels - rules/ai Life prediction -physics/terils odels Fuzzy logic Probbilistic echnics Optiiztion Finncil nlysis Tble. - Key Eleents in the Prognosis Process. As indicted in tble. [3], the fundentl gol of ll of these pproches is to fcilitte decisions bsed on better infortion whether for ission plnning in the field (over the short ter), or sustinent t the depot (over the longer ter). In fct, the optiu prognosis syste is likely to be soe cobintion of trditionl dt-driven ethods nd probbilistic echnics ethods. Thus, in ny respects the bove tools cn be viewed s being copleentry. 6
I.4 - Prognostic Definition The ter prognostic founds its origin in the Greek word progignôskein which ens to know in dvnce. Industril Prognostic is clled the prediction of syste s lifetie nd corresponds to the lst level of the clssifiction of dge detection ethods introduced by []. Prognostic cn lso be defined s probbility esure: wy to quntify the chnce tht chine opertes without fult or filure up to soe future tie. This "probbilistic prognostic vlue" is ll the ore n interesting indiction s the fult or filure cn hve ctstrophic consequences (e.g. nucler power plnt) nd intennce nger need to know if inspection intervls re pproprite. However, sll nuber of ppers ddress this connottion for prognostic [4,5]. Finlly, lthough there re soe divergences in literture, prognostic cn be defined s: "prognostic is the estition of tie to filure nd risk for one or ore existing nd future filure odes" [6]. In this connottion, prognostic is lso clled the "prediction of syste's lifetie" s it is process whose objective is to predict the reining useful life (RUL) before filure occurs given the current chine condition nd pst opertion profile [7]. The in steps defined in this stndrd re surized in figure.5. Monitoring Dignostic Prediction Posterior ctions Figure.5 - Sury of the ISO 338-: 4 Stndrd Min Steps The first step consists of onitoring the syste by set of sensors or inspections chieved by opertors. The onitored dt re then pre-processed in order to be used by the Dignostic odule. The output of this odule identifies the ctul operting ode. This stte is then projected in the future, by using dequte tools, in order to predict the syste s future stte. The intersection point between the vlue of ech projected preter or feture nd its corresponding lr threshold leds to wht is known s RUL (Reining Useful Life) of the syste (figure.6). Finlly, pproprite intennce ctions cn be tken depending on the estited RUL. These ctions y i t eliinting the origin of filure which cn led the syste to evolve to ny criticl filure ode, delying the instnt of filure by soe intennce ctions or siply stopping the syste if this is judged necessry. 7
Figure.6 - RUL Intervl Definition. I.5 - The Role of Prognostic in Lifetie Process Ech syste or coponent of syste psses by three periods during its functioning life. The lst phse during ech syste life represents the degrdtion period leding to filure by progressive deteriortion. It is iportnt to predict, t ech instnt, the reining lifetie in order to prevent expensive defects nd to void ctstrophic filures. Prognostic is process encopssing cpcity of prediction. It is the bility to estite the reining useful lifetie (RUL) of equipent in ters of its functioning history nd its future usge. Predicting the RUL of industril systes becoes currently n iportnt i for industrilists knowing tht the filure, whose consequences re generlly very expensive, cn occur suddenly. The clssicl strtegies of intennce [8] bsed on sttic threshold of lr re no ore efficient nd prcticl becuse they do not tke into considertion the instntneous product functioning stte. Adopting preventive systetic intennce by frequent replceent to increse the syste vilbility is n expensive strtegy [8,9]. The introduction of prognostic pproch s n "intelligent" intennce consists of the nlysis, the helth follow up nd onitoring, bsed on physicl esureents using sensors. 8
The RUL of syste in service cn be expressed in hours of functionning, in Kiloeters run or in cycles. If we cn effectively predict the condition of chines nd systes, intennce ctions cn be tken hed of tie. Good nd relible prognosis needs good nd relible dignosis. The science nd technology of prognosis nd structurl helth ngeent offer the potentil for significnt enhnceents in the sfety, relibility nd vilbility of high-vlue resources [3,3]. This concept is bsed on closed-loop process whose successful ipleenttion depends on the integrtion of severl ulti-disciplinry eleents including [3]: ) Onbord sensing of opertionl preters nd teril dge sttes; ) Dignosing trends, fult conditions, nd underlying dge; 3) Predicting reining useful life in ters of probbility of filure nd liits on relible perfornce, 4) And deciding upon pproprite courses of ction: whenever or not the resource is cpble of perforing given ission, or lterntively, is in need of inspection, intennce, or replceent. I.6 - Stte-of-the-Art of the Prognostic Approches Vrious pproches to prognostics hve been developed tht rnge in fidelity fro siple historicl filure rte odels to high-fidelity physics-bsed odels [3]. The required infortion (depending on the type of prognostics pproch) include: engineering odel nd dt, filure history, pst operting conditions, current conditions, identified fult ptterns, trnsitionl filure trjectories, intennce history, syste degrdtion nd filure odes. Putting t work prognostic process consists of executing set of tretent fro input infortion. The different pproches of prognostic re grouped in function of their pplicbility s well s their econoic yield. They re three filies [,3]: - The pproches bsed on odels (Model-bsed prognostics) - The pproches guided by dt (Evolutionry or trending odels) - The pproches bsed on experience (Experience-bsed prognostics) 9
The pyrid reproduced in the figure.7 highlights the hierrchy of these different filies. According to [33], king the choice of n pproch fily is done by nswering two questions: - Is it possible to construct physicl odel for the degrdtion echniss? - Is the instruenttion of the equipent sufficient in order to evlute degrdtion evolution indictor? If the nswer to the first question is positive, the ipleenttion of n pproch bsed on physicl odels is considered. Moreover, if the nswer to the second question is positive, n pproch guided by dt is possible. In the cse where the nswer to the two questions is negtive then n pproch bsed on expert knowledge nd feedbck is the best solution. A study relized on ore thn publictions in the field of prognostic [34] shows tht in the industril sector, the pproches guided by dt nd bsed on experience re the ost ipleented ones. - Physicl Models - Avilble Sensor Model-Bsed Procedure - Estition Models - Avilble Sensors Model-Bsed Prognostic Pttern Recognition, Fuzzy Logic, eurl etwork Estition-Bsed or Trending Prognostic - Relibility Models, Sttisticl Models - o Avilble Sensors Experienced-Bsed Prognostic Signl Processing Algorith Rnge of Syste Applicbility for Ech Prognostic Method Figure.7 - Prognostic Technicl Approches.
I.6. - Prognostic Bsed on Models This pproch is lso clled odel-driven or physicl odel. As its ne indictes, this pproch fily uses odels tht cn be of two different types [35,]: - Model bsed on the equipents physics - Mtheticl odels constructed by experienttion This "Physicl odel" is bsed on theticl description of degrdtion process nd on its level evolution using DI onitoring (on-destructive Inspection). It is described to be ore flexible nd precise thn the two other pproches. The degrdtion is then considered s continuous vrible whose evolution is chrcterized by deterinistic or stochstic lw. The concept of these ethodologies is to ke the constructed odel evolve till wnted future instnt, fro n initil degrdtion stte nd the future usge of the equipent [36]. The equipent is considered s fulty when the degrdtion vrible reches predefined threshold in the cse of n isotropic odel, or predefined surfce in the cse of non-isotropic odel. These odels cn be: nonliner equtions [37], odels defined by expert nlysis [38], or even by physicl odels of cheicl corrosion [39], of echnicl ftigue [4], etc. For soe equipents nd criticl structures, it is necessry to estite the initition nd the crck propgtion. The odels bsed on crck propgtion re interested in the probles deling with teril properties, nd they hve evidently n iportnt interest in prognostic, but they re usully dpted for rel-tie tretent due to their big coputtionl coplexity [8]. A technique, ong others, cpble of predicting the slope of increse nd the directions of the crck, is the siultion by decoposition in finite eleents. The decoposition in finite eleents is used to study the behvior of n equipent in different disciplines such s therodynics, fluids echnics, structures echnics etc...the ethod of finite eleents is bsed on the ide tht coplex syste cn be subdivided into sll prts clled eleents. Ech eleent is copletely defined by its geoetry nd its physicl properties. The study of ech eleent is then sipler thn the study of the coplete structure tht they copose. Ech eleent cn be considered s continuous prt of the structure. The decoposition in finite eleents converts continuous structure into syste of lgebric equtions or into ordinry differentil equtions. The solution of
proble using the theory of finite eleents invokes ethods of reserch of siultneous solutions to the rection of ech eleent to chrges, to constrints, nd to the interction ong the djcent eleents. An exple of the ppliction of this theory is the prognostic for syste of trnsission of helicopter; it is presented in [4]. The odel-bsed ethods ssue tht n ccurte theticl odel cn be constructed fro first principles. This pproch to prognostic requires specific filure echnis knowledge nd theory relevnt to the onitored chine. The existing ppers propose different odel bsed solution for the industril probles. Brtelus nd Ziroz [4] estited through deodultion process, the vibrtion signl for plnetry gerbox in good nd bd conditions. Kcprzynski et l. [43] proposed fusing the physics of filure odeling with relevnt dignostic infortion for helicopter ger prognostic. helidze nd usuno [44] proposed generl ethod for trcking the evolution of hidden dge process given sitution where slowly evolving dge process is relted to fst, directly observble dynic syste. Luo et l. [45] introduced n integrted prognostic process bsed on dt fro odel-bsed siultions under noinl nd degrded conditions. Oppenheier nd Lopro [46] pplied physicl odel for predicting the chine condition in cobintion with fult strengths-to-life odel, bsed on crck growth lw, to estite the RUL. Ads [37] proposed to odel dge ccuultion in structurl dynic syste s first/second order nonliner differentil equtions. helidze [47] odeled degrdtion s "slow-tie" process, which is coupled with "fst-lie", observble subsyste. The odel ws used to trck bttery degrdtion (voltge) of vibrting be syste. Li et l. [48] nd [49] introduced two defect propgtion odels vi filure echnis odeling for RUL estition of berings. Ry nd Tngirl [5] used nonliner stochstic odel of ftigue crck dynics for rel-tie coputtion of the tiedependent dge rte nd ccuultion in echnicl structures. A different wy of pplying odel-bsed pproches to prognostic is to derive the explicit reltionship between the condition vribles nd the lifeties (current lifetie nd filure lifetie) vi filure echnis odeling. Two exples of reserch long this line re [5] for chines considered s energy processors subject to vibrtion onitoring nd [5] for berings with vibrtion onitoring. In [53] nd [54] the proble of forecsting chine filure is illustrted for high power fn bering nd rilrod diesel engine. Engel et l. [8] discussed soe
prcticl issues regrding ccurcy, precision nd confidence of the RUL estites. Lesieutre et l. [55] developed hierrchicl odeling pproch for syste siultion to ssess the RUL. A first exple is given by helidze who odels the loss of tension (degrdtion) of bttery providing energy to n electrognetic oscilltor, by coupling two odels [56,57]: x f φ ε ( x, µ ( φ), t) g( φ, x, t) () where x is n observble vrible of the syste stte, φ is n internl sclr vrible relted to the degrdtion, ε represents the difference in tie scle, < ε <<. µ (φ ) represents the vrition of the bttery chrcteristics in function of the degrdtion. Moreover, Kln ~ filter is used to deterine the current vlue φ ( t ) in function of the observed esures. The estition of the Tie To Filure (TTF) denoted by T TTF is then given by the solution of the eqution [58]: φ g(φ) () where g is obtined by pplying the concept of ens to g. The odel of degrdtion used for prognostic is then relted to the originl slow subsyste () by tking the en on long period of the field of vectors of g, hence the tie to filure will be: dφ g( φ) φ T liit TTF ~ (3) φ ( t) with φ liit is the criticl vlue of degrdtion for which the bttery is considered s unusble. A second exple is the proposition of the generic ethodology in the cse of odels with n ppliction to qurter of vehicle suspension [59]. This used odel is very close to the previous one: x ( x, λ( θ ), u, t) f θ ε g( x, θ, t) y x + Du + v (4) 3
where x is the syste stte vector, θ is the degrdtion stte vector, λ is the syste preter vector in function of the degrdtion stte, u is the syste input vector, ε is the tie scle, y is the syste output vector, nd v is the esure noise..8 below: The generic ethodology proposed for odel bsed prognostic is reproduced in figure Syste Modeling Preters Rndo Siultion Degrdtion Estition Prognostic Model Degrdtion Follow-Up Prediction of the RUL Figure.8 - Generic Methodology for the Model-Bsed Prognostic According to [6] The first step consists of estblishing odel using coupled differentil equtions (4). The second step is the siultion of the odel obtined under different operting conditions. The input vector u is n uncertin eleent corresponding to inputted loding or excittion. It is then necessry to identify the different operting odes (the different clsses of the input 4
vector) whose preters re defined by the lws of probbility. A Monte rlo siultion is then executed for ech operting ode. The result is set of degrdtion trjectories. During the siultion, in order to decouple the slow-tie ode fro the fst-tie ode, the principle of the en is used. Tht ens tht the stte of degrdtion is coputed t fixed period before injecting it in the fst-tie ode. The different trjectories obtined for the different functioning odes define the prognostic odel. The degrdtion estition step consists of defining ethod of degrdtion observtion or n ige of degrdtion fro the syste esure vector y. The follow up step of degrdtion llows on one hnd to deterine the current vlue of the degrdtion stte nd on the other hnd to construct prediction odel of the operting odes by using tool such s Mrkov odels. To finish, the prognostic is relized by projecting the degrdtion trjectory following the prediction odel of the functioning ode estblished in the previous step, until the stte of degrdtion reches the liit threshold φ liit. The estition of the degrdtion stte is key point in the success of the ethodology but it reins very difficult due to the fct of the very wek degrdtion dynics nd due to the esureents noises. A ethod bsed on the use of observers of convergence in finite tie in order to estite the stte of degrdtion of odel siilr to (4), is presented in [6]. I.6.. - Advntges nd Drwbcks of the First Approch The in dvntge of odel bsed pproches is their bility to incorporte physicl understnding of the onitored syste [58]. In ddition, in ny situtions, the chnges in feture vector re closely relted to odel preters nd functionl pping between the drifting preters nd the selected prognostic fetures cn be estblished [58]. Moreover, if the understnding of the syste degrdtion iproves, the odel cn be dpted to increse its ccurcy nd to ddress subtle perfornce probles. onsequently, they cn significntly outperfor dt-driven pproches (next section). But, this closed reltion with theticl odel y lso be strong wekness: it cn be difficult, even ipossible to ctch the syste's behvior. Further, soe uthors think tht the onitoring nd prognostic tools ust evolve s the syste does. 5
I.6. - Prognostic Guided by Dt This pproch is lso clled Dt-driven or evolutionry or trending or estitionbsed pproch or rtificil intelligence. In certin cses, it hppens tht we dispose of dtbse contining the history of scenrio degrdtion/filure represented by set of tie series. These bses re given without the use of physicl odel of equipent behvior. The evolution of the degrdtion indictor is then relized with the help of sttisticl ethod. Depending on the ethod used, three clsses of pproches cn be distinguished [3,6]: - The prognostic by trend nlysis - The prognostic by lerning - The prognostic by stte estition The indictor or the indictors of degrdtion re priordil eleents of prognostic driven by dt. They re deterined by sttisticl clcultion tht quntifies the stte of the equipent wer out. The ulti-vribles sttisticl techniques re powerful tools cpble of copressing dt nd reducing their diensions in wy tht the essentil infortion is intined. They cn lso nipulte the noise nd the correltion in order to extrct infortion efficiently. The principle function of this type of techniques is, using theticl procedure, to trnsfor certin nuber of correlted vribles into sller set of non-correlted vribles [63]. The dt-bsed pproches require tht the infortion extrcted fro sensors be sufficient in qulity nd quntity in order to evlute the current stte or the ige of the current stte of the syste degrdtion. The concept of this pproch consists of collecting infortion nd dt fro the syste nd projecting the in order to predict the future evolution of soe preters, descriptors or fetures, nd thus, predict the possible probble fults. Without being exhustive, theticl tools used in this pproch re inly those used by the rtificil intelligence counity, nely: teporl prediction series, trend nlysis techniques, neuronl networks, neuro-fuzzy systes, hidden Mrkov odels nd dynic Byesin networks [4,7,6]. 6
The dvntge of this pproch is tht, for well onitored syste, it is possible to predict the future evolution of degrdtion without ny need of prior theticl odel of the degrdtion. However, the results obtined by this pproch suffer fro precision, nd re soeties considered s locl ones (for the cse of neurl networks nd neuro-fuzzy ethods). In ddition, the onitoring syste ust be well designed to insure cceptble prognostic results. The Dt-driven pproches use rel dt gthered on-line with sensors or by opertor esures to pproxite nd trck fetures reveling the degrdtion of coponents nd to forecst the globl behvior of syste. Indeed, in ny pplictions, esured input/output dt is the jor source for deeper understnding of the syste degrdtion. Dt-driven pproches cn be divided into two ctegories: rtificil intelligence (AI) techniques (neurl networks, fuzzy systes, decision trees, etc.), nd sttisticl techniques (ultivrite sttisticl ethods, liner nd qudrtic discriintors, prtil lest squres, etc.) [4,7,6]. I.6.. - Prognostic by Trend Anlysis This type of pproch is bsed on the derivtion of the indictor of the degrdtion stte fro its norl functioning stte. The tools used in order to put in work these pproches re the tools of prediction of tie series nd the odels of ulti-vribles clssifiction. The choice of tool depends on the nuber of degrdtion indictors s well s on the nuber of odes of functioning identified. The tool y be very siple like for exple liner regression. In this cse, the n lst points coputed fro the degrdtion indictor re used to estite the coefficients of the ffine function chrcterizing the indictor trend. Prognostic is then ccoplished by the deterintion of the point of intersection of this function with the criticl threshold of filure. The result of prognostic is then in this cse, the tie before equipent filure [44]. Bsed on the se principle, predictive odel of type ARMA (Auto Regressive with Mobile Averge) cn be used [64]. The preters of this odel re then updted in rel tie with the help of lest squres lgorith. The uthors in [65] use prediction ethod for the degrdtion stte of copressor. The tool used for this type of prognostic could be the Principle oponents Anlysis technique (PA) or the liner nd qudrtic discriintion [66]. These tools cn be lso pplied on teporl indictors or on frequency indictors [67]. 7
Generlly, this type of prognostic gives better results t the syste level rther thn t the equipent level since the syste perfornces degrdtion is usully the result of the interction of the different constituent equipents with degrded functioning [44]. The trend nlysis nd the indictor prevision cn be lso relized in function of the vribles influencing the degrdtion [68]. I.6.. - Prognostic by Lerning This type of prognostic uses principlly techniques issued fro chine lerning nd rtificil intelligence. urrently, the principle techniques used re Artificil eurl etworks (A) [69]. An A is tool, generlly used for nonliner odels, tht llows estblishing functionl reltion between n inputs vector nd desired outputs vector. The preters of these odels re djusted in order to hve optil perfornces. Different techniques cn be used to djust these preters such s the optiiztion technique. The network is, firstly, trined by using dt representing the evolution of degrdtion during the whole equipent lifetie, until filure occurs. Afterwrd, the network is used to detect or predict n evolution of the degrdtion indictor using other dt, lwys reining in the se odes of functioning during the period of lerning. The inputs of the network re generlly the discrete vlues of the indictors fro instnt tk n till t k nd the outputs re: - Either the current stte of the equipent. In this cse the network relizes clssifiction in order to know the input sitution bsed on the lerned situtions. - Or either the vlues of the degrdtion indictors t instnt t k + T relizes then n extrpoltion fro the input sitution.. The network In the doin of A, the Dynic Wvelet eurl etworks (DW) re used. Their structure is of the for: y ( y,, y, u, u ) W, k + k k k (5) k n with uk is the input vector nd y k is the output vector, nd n s being the nuber of inputs nd outputs history vectors nd which re kept in eory. 8
W is neurl network with sttic wvelets. It is then recursive odel tht links in dynic wy ctul, old, nd future dt. This type of networks cn be trined in function of tie, by using lgoriths which cn be dvnced ones such s the genetic lgoriths [7], [7]. One of the principl dvntges of this kind of networks is tht the input vector cn be de out of signls of different kinds nd even of ixture of teporl nd frequency signls. This network ws used for prognostic fro the vibrtions signls of rotting chine [7] nd lso for the prediction of crck evolution in copressor. Other fors of A cn be lso used [73,74] such s the recurrent networks of rdil functions. An ppliction for the prognostic of gs oven is presented in [75]. A cse study on the prognostic of the filure of the opening door syste in n irport bus is described in [76]. Since few yers, other techniques such s the Relevnce Vector Mchine (RVM) lgorith hve been used [77]. It llows the construction of probbilistic odel of Byesin for representing the generlized liner odel in for of function identicl to the lgorith of Support Vectors Mchine (SVM). The lgorith RVM considers set of n given dt { x i, y i } with i [,n] nd with x vector of diension q ssocited with y i. The lgorith ws initilly defined in order to deterine the probbility ( y x) ~ ( f ( x), ) p where is the vrince of the noise dded to the dt. The principle of the lgorith is to guess the underlying probbility distribution tht genertes the dt: ( y ω, ) exp y φ p ω (6) π where φ is trix contining the nucleus. The prediction function obtined then is of the for: f n ( x ) φ( x ) i ω, + ω (7) i x i with ω i s the weights ssocited with ech support [78]. The key concept of the lgorith RVM for prognostic is its probbilistic interprettion of the output y. 9
Other techniques like fuzzy logic cn be eqully used to copleent the tools of chine lerning for the prognostic of lerning [79]. Fuzzy logic, prticulrly, llows the use of linguistic vribles in the dynic odel in order to tret uncertinty tht lies t the hert of the perfornce of prognostic lgorith [8]. Within the field of intennce probles, Artificil eurl etworks (As) nd neuro-fuzzy systes (F) hve successfully been used to support the detection, dignostic nd prediction processes, nd reserch works ephsize the interest of using it [7,8,8,83,84]: As nd Fs re generl nd flexible odeling tool, especilly for prediction probles. I.6..3 - Prognostic by Stte Estition The pproch by stte estition is usully used when onitoring syste by iges nd pttern recognition is lredy put t work on the equipent [85]. The for is, in this cse, considered like n ige of the equipent degrdtion. The gol of prognostic is then to predict the for evolution. Prognostic by stte estition ssues tht the degrdtion evolution cn be expressed by the following stochstic for of discrete tie [8]: x z k f ( xk, ωk ) p( xk xk ) g ( x, ν ) p( z x ) k k k k k k k (8) where x k is vector contining the degrdtion stte, nd ω k nd ν k re the preters of the environent tht influence the evolution of the degrdtion, they re non-gussin noises, f k nd g k re functions, nd zk is vector of degrdtion stte. Like in the other prognostic pproches, the first step consists first of ll in estiting the current vector x k, nd then prognostic is done. Two cses re possible depending on the for of functions f k nd g k. In the cse where f k nd g k re such tht: f k ( xk ωk ) Ak g ( x, ν ), k k xk + ωk x k k (9) 3
where A k is trix contining the odel trnsition preters, it is possible to predict the evolution of the sequence { ˆ }, i [, n] x k + i, fro the sequence of observtions { xˆ }, j [,k] j. This technique ws pplied on engines of continuous currents [86] nd on ger systes when cobined to fuzzy logic [87]. If now f k nd g k re nonliner functions, it is possible to use ethod bsed on prticulr filtering [8] tht seeks to reove noise, to reduce dt size by copression, nd to sooth the resulting tie series in order to identify their generl ptterns (velocity, ccelertion, etc.), nd this by using typicl signl-processing lgoriths (edin filter nd rectngulr filter). The estition of the current stte is then given by the knowledge of process odel nd by the estition of the previous stte: p ( xk z,, k ) p( xk xk ) p( xk z,, k ) dxk () The prediction of the degrdtion evolution fro the estition of the current stte on horizon q is given then by: p k + q ( xk q z,, k ) p( xk z, k ) p( x j z j ) dxk,, k + p () + j k + An exple of fult nticiption with the help of prticulr filtering is syste coposed of three curves nd presented in [88]. Using the se principle, n ppliction of tie prediction before filure of syste hving crck, is chieved in [89]. I.6..4 - Advntges nd Drwbcks of the Second Approch The strength of dt-driven techniques is their bility to trnsfor high-diensionl noisy dt into lower diensionl infortion for dignostic/prognostic decisions. AI techniques hve been incresingly pplied to chine prognostic nd hve shown iproved perfornces over conventionl pproches. In prctice however, it isn't esy to pply AI techniques due to the lck of efficient procedures to obtin trining dt nd specific knowledge. So fr, ost of the pplictions in 3
the literture just use experientl dt for odel trining. Thus, dt-driven pproches re highly-dependent on the quntity nd qulity of syste opertionl dt. I.6.3 - Prognostic Bsed on Experience This pproch is clled experience bsed or probbility bsed or sttisticl bsed prognostic pproch. It is necessry where we cnnot use the two previous pproches. It is bsed on relibility function or on Byesin process where the preters re tken fro feedbck experience or expert opinion. Its disdvntges re the incpcity to tret coplex systes of ny coponents nd its exclusive binry principle (success/filure) rther thn continuous sttes of degrdtion. When obtining physicl odel of n equipent is difficult nd it is ipossible to estite degrdtion fro the sensors instlled on the equipent, prognostic bsed on experience cn be the only lterntive [3]. This for of prognostic is the less coplex but requires n excellent feedbck fro experts in for of historicl dt, of knowledge bse or of expert dt. This expertise llows stochstic or probbilistic odeling of degrdtion. It is the for the best dpted to coplex systes tht re very difficult to odel physiclly nd whose degrdtion indictors re sensitive to usge conditions [33]. This prognostic pproch consists of using probbilistic or stochstic odels of the degrdtion phenoenon, or of the life cycle of the coponents, by tking into ccount the dt nd knowledge ccuulted by experience during the whole exploittion period of the industril syste. The probbilistic odel cn be siple probbility function or odeling in the for of stochstic process. In this frework, the ost used probbility functions re: Weibull lw, exponentil lw when the filure rte is supposed to be constnt, nd norl, log-norl nd Poisson lws. The preters of ech lw re estited fro the dt gthered during the whole exploittion period of tie (experience feedbck, intennce dt, etc.). Stochstic process odels cn be Mrkovin or sei-mrkovin. 3 The experience-bsed odels [6] re bsed on esureents tken fro helth onitoring of chine like for exple those bsed on expert judgent, stochstic odel,
Mrkovin process, Byesin pproch, Relibility nlysis, Optiiztion of preventive intennce, etc.). Their prognostic ethodology proves to be siple but inflexible towrd chnges in syste behvior nd environent. I.6.3. - Stochstic Approch This type of pproch is chrcterized by odeling the equipent life by stochstic degrdtion process. The jor prt of the works in this field represents the degrdtion process by Mrkovin or sei-mrkovin odels [9,9]. The equipent psses then through different sttes of degrdtion. Prognostic consists of deterining either the reining useful lifetie, or the probble future stte or sttes of the equipent in function of its current stte if the process used is Mrkovin or in function of its stte nd of tie spent in this stte if the process is sei-mrkovin. Figure.9 illustrtes sei-mrkovin process. The set { d j } j [, n],, represents the different degrdtion sttes: d no degrdtion,..., d n xil degrdtion. The p i, j represent the trnsitions probbilities fro stte d i to stte of the equipent T ν is given by: d j. The reining useful lifetie n j ( ) Tν D d j () with D ( d j ) is the durtion ssocited with the stte d j. The prognostic lgorith used is the following [9]: - Obtin the trnsition probbilities trix fro lerning procedure. - Deterine the probbility densities of the durtion of ech stte d j. - Identify the current stte d k of the equipent. - lculte the current reining useful lifetie RUL k fro the reining useful lifetie RUL k + t next instnt in ters of the trnsition probbility between the two instnts nd the self-stte probbility. ( D( dk ) + RULk + ) + pk, k + RUL RUL (3) k pk, k k + 33
p p,, 3, 3 p p n, n p p,, 3 3, 4 d d d3 p......... p n, n d n Figure.9 - Prognostic Bsed on Hidden Sei-Mrkovin Process. The use of the sei-mrkovin odel is preferble copred to the Mrkovin odel since, the ltter, ssues tht the chrcteristics of the degrdtion process cnnot be odified grdully with tie. Moreover, the previous odels re insufficient in order to odel the degrdtion process tht tkes into considertion fctors of influence linked to the environent or the equipent use. To do so, it is necessry to use odels of stte chnge tht tke into considertion the influence of these fctors. The stte of these fctors odifies the vlue of the evolution preters of the degrdtion process odel [33]. I.6.3. - Relibility Approch This pproch is bsed on probbilistic odeling of the filure instnt, of the equipent relibility. The relibility of n equipent group t n instnt t is the probbility of operting without filure during the period [,t]. Although it is represented by teporl for, this definition reins vlid with other units such s the kiloeter or even the nuber of cycles of opertion. The relibility function R (t) of n equipent is deterined fro lrge popultion of the se equipent. It is coputed by: uber of eleents in life t the instnt t R ( t), t (4) Totl nuber of eleents The function R (t) llows, then, to define f (t), the probbility density of the vrible T which represents the filure instnt. The function tht the filure instnt T is between t nd t + dt. f ( t) dt chrcterizes thus the probbility dr( t) f ( t), t (5) dt 34
There exists ny stndrd distribution functions tht llow to odel f (t). The ostly used is the Weibull distribution: Weibull β β t γ t γ ( t, β, η, γ ) exp η η β η (6) Where: β is the for or shpe preter, η the scle preter, nd γ the shifting preter function of tie or loction. We note tht the curves of the Weibull distribution chnge in shpe considerbly for different vlues of the preters, prticulrly the preter β. If β, The Weibull distribution reduces to the exponentil distribution. For vlues of β > the curves becoe soewht bell-shped nd reseble the orl curves but disply soe skewness. Other distributions re eqully used such s: the Poisson lw or the Binoil lw, the norl lw, the exponentil lw, the g lw, etc... In the relibility pproches, prognostic is chieved with the help of the rte of filure λ (t) tht defines the conditionl probbility of the occurrence of filure t instnt t given tht the device survived until instnt t-. In the cse of Weibull distribution, λ(t) is s follows: β f ( t) β t γ λ ( t), t (7) R( t) η η Experientl observtion shows tht λ (t) hs the for of curve sid bthtub curve reproduced in figure.. The evolution of λ (t) is generlly decoposed into three periods: - Youth sybolizes the precocious filures, in the cse of Weibull lw: β <, - Exploittion where the filure rte is lost constnt, β, - End of life, wer-out, where we observe the occurrence of filures, β >. 35
Figure. - The Bthtub urve of Filure Rte versus Tie. In rel usge conditions, relibility nd degrdtion of n equipent re influenced by two sets of preters [93]: - The environent (teperture, huidity, etc), - The ode of functioning (work lod, stte, etc). In odeling point of view, the introduction of vector z (t) perits to tke into considertion these two sets of preters in the expression of R (t) or λ (t). In the first cse, the deteriortion process R (t) is ccelerted. We spek hence of n Accelerted Life Model (ALM) [94]: R( t) R ( t) ( z( t) ) Ψ t, (8) In the second cse, the rte of filure λ (t) increses in function of usge conditions. We spek thus of Proportionl Hzrd Model (PHM) [95]: ( z(t) ) ( z( t) ) λ ( t), λ ( t) Ψ t (9) Ψ is function of the vector z (t). It represents the physicl behvior tht governs the degrdtion in ters of the environent nd the ode of functioning of the equipent. R ( ) t 36
nd λ ( ) re respectively the relibility nd the rte of filure in the noinl usge t conditions. In the doin of the prediction of the relibility of electronic systes, consortiu of eight industrils of defense eronutics, hs developed new ethodology clled FIDES (Fonds D'investisseent pour le Développeent Éconoique et Socil) [96]. This ethodology is bsed on tking into considertion three coponents: Technology, Process, nd Usge. The usge considers the equipent eployent constrints through the profile of the ission, by subdividing the ission into phses into which the constrints re constnt. The objective of the FIDES odels is to llow relistic evlution of the electronic equipents relibility including for the equipents tht encounter severe environents. The generl odel is of the for [97]: λ () equipent Physicl ontributions Process ontributions where the ter Physicl ontributions is n dditive ter tht represents the physicl nd technologicl contribution to relibility such s: the type of terils used in the equipent construction. The ter Process ontributions is ultiplictive ter tht represents the ipct of the developent process, of production nd exploittion on relibility. This ethodology gve birth to guiding nul contining, for ech electronic equipent, tbles of the different fctors tht contribute to relibility. I.6.3.3 - Advntges nd Drwbcks of the Third Approch The dvntge of the ethods of this pproch is tht it is not necessry to hve coplex theticl odels to do prognostic. Moreover, this pproch is esy to pply on systes for which significnt dt re stored in se stndrd tht fcilittes their use. For exple, copny which hs built during long period of tie production nd intennce dtbse with soe inor rules nd stndrds for dt storing, cn esily get the estition of the preters of the probbility lws. However, the in drwbck of this pproch dwells in the ount of dt needed to estite the preters of the used lws. Indeed, huge nd significnt ount of exploittion dt re needed in order to deterine preters tht odel fithfully the degrdtion phenoenon or the life cycle of the concerned syste. onsequently, this pproch cnnot be 37
pplied in the cse of new systes for which dt fro experience feedbck do not exist. The other kind of proble is tht in ost cses, it is necessry to filter nd pre-process the dt to extrct the useful ones, becuse the stored dt re not lwys directly exploitble (for exple, in the se copny, two intennce opertors y enter in two different infortion or pprecitions for the se resolved proble). I.6.4 - Methodology Bsed on Abci of Degrdtion Severl prognostic studies re proposed nd re bsed on bci of degrdtion under clss of incresing functions without ny nlytic for like in the work of Peysson et l. [98]. Their pproch belongs to the Dt-driven fily of prognostic pproches. The prognosis work of Peysson et l. on vehicle suspension syste ws bsed on the bcus of degrdtion under clss function F. We know tht these functions re incresing. Figure. shows three odes of degrdtion reltive to the three sttes of the rod (very good condition in red, fir condition in blue, nd severe condition in green). Degrdtion esure D Degrdtion esure D Opertionl Tie ( 5 s) Opertionl Tie ( 5 s) Figure. - The Three Modes of Degrdtion. Figure. - The Modelistion of the Abci of Degrdtion.,,,3 The degrdtion set DD r is given by: { ( τ ), ( τ ), ( τ )} D (figure.). To obtin this set, the vlues of the following preters ust be clculted by: r α i, k i, k Ln y β Ln x exp b i, k e η y b i, k ( Ln y β Ln x ) () Ln y Ln x eb y b b b 38
Where, x : the operting tie y : the degrdtion stte, with: y < y e : the devition fro extree points : b y + e ; y e The results of the three odelistions for bci of degrdtion re indicted in [98]. The vlues of the triplets x, y, e ) nd ( x, y, e ) for the three odes re clled the ( b b b bci coefficients nd re indicted in [98]. For the unique utiliztion profile u,, the environentl vrible (stte of the rod) is de discrete into three context conditions {, c c } c, shown in [98]., 3 To nlyze the trjectory of degrdtion of the resources, we tke here the suspension s the only resource RR, we consider society of brke delivery equipped with two identicl vehicles: veh nd veh. They ke weekly ission of the se durtion (35 h) nd of the se distnce but with different rod qulity. They coplete the se ission MM but they re subject to different environentl constrints (rod stte). The environentl sequences encountered by the two vehicles re respectively nd. The durtion is expressed in hours by: (( c3, ), ( c,),( c,6),( c,4),( c3,3) ) (( c,),( c,7), ( c,6), ( c,3), ( c,9)) 3 3 () The nlysis of degrdtion trjectory reltive to the suspension resource of the two vehicles llows to better pln the intennce of ech vehicle in order to prevent filure nd to increse the profitbility. To estite the tie before suspensions filure, then the lgorith is executed while D < (no filure cse). The uthors deduce in [98] the bci curves nd the degrdtion trjectories. According to the ethodology of Peysson et l., the curve of ech trjectory is given in ters of its use profile in function of the environentl context. The odel cn be siply odified to dd soe dynics of degrdtion. The relized prevision llows us to deterine the success of the ission. 39
This proposed pproch cn be treted by three different wys: - Firstly, before the ission, the nlysis of the trjectory identifies the defective resources nd it gives lso the pproxite vilbility tie. - Secondly, fter the ission, the necessry vribles for the operting odel re registered nd stored during the ission in order to be treted, to nlyze subsequently the degrdtion trjectories, nd to know the ission ipct on the syste degrdtion. - Thirdly, during the ission, this wy is the intersection between the two previous utiliztions. The use during the ission llows redjusting the prevision in rel tie. The follow up of the degrdtion trjectories in rel tie nd the correction of previsions cn lso be used s tools tht help in decision king to iniize the resources of degrdtion. This ethodology cn lso be used s tool to understnd the behvior of coplex systes, in order to void strong degrdtions. Peysson et l. hve concluded with n exple of ppliction using the GPS dt. If we ssue tht the crtogrphy GPS includes dt on the stte of the rods, then the GPS disposes eteorologicl previsions nd is connected to the vehicle sensor. As the ethodology of Peysson et l. is essentilly bsed on expert systes, it is relying on the sttistics of lrge esured dt (s exples we cn cite the works bsed on degrdtion behvior described by bci nd using expert description of syste: Process- Mission-Environent [], the works bsed on rtificil intelligence, chine lerning [99], neurl network [6], fuzzy logic [], etc.). Their ethodology bsed on bci of degrdtion belongs to the Dt-driven fily of prognostic pproches. It is useful in ny rel cses (like the ship exple where ny internl nd externl preters influence its ission). It is dequte when huge nuber of dt is necessry to be included into the prognostic process. I.7 - Sury We hve presented in the previous sections stte of the rt of the different pproches invented nd pplied for prognostic function. Tble. presents coprtive sury for need of prognostic in the cse of three filies of pproches. We note tht the 4
jor prt of the presented pproches pply for n eleentry coponent of prognostic nd rein difficult to use for coplex syste. In the pproches bsed on physicl or theticl odels, the knowledge of the equtions of the dynic behvior of degrdtion kes their use very flexible. In cse when the syste properties or of degrdtion chnge, then the odel preters cn be redjusted. But the developent of such odel is very expensive becuse it is necessry to hve high level of qulifiction in order to ster the echniss of equipent degrdtion. This type of odel lso presents coputtionl difficulties during its siultion. Prognostic ccurcy Experience-Bsed Evolutionry Physics-Bsed Engineering ot required Beneficil Required odel Filure history Required ot required Beneficil Pst operting Beneficil ot required Required conditions urrent Beneficil Required Required conditions Identified fult ot required Required Required pttern Mintennce Beneficil ot required Beneficil history In generl o sensors/no odel Sensors/no odel Sensors nd odel Tble. - Sury of the Three Prognostic Approches [6] The pproches guided by dt ssue relible estition of the stte or the ige of the current stte of degrdtion in order to predict the future evolution of the syste. The ethods of trend nlysis lck rectivity when fcing chnge in usge conditions. The efficiency of the lerning ethods is strongly linked to the spling of dt tht serves to copute the odel preters. If n unlerned sitution occurs, prognostic cn be rndo. These ethods bsed on stte estition require odel of the degrdtion indictor behvior nd they re sensitive to the operting ode. Experience bsed prognostic, either the stochstic pproch or the relibility pproch, requires little expert knowledge of the degrdtion echniss. It reins esy to ipleent but it is not rective when fcing chnge in the syste operting ode. In fct, the odels usully creted nd devised re considered s verge odels of ny equipents. Although 4
ny solutions were found in order to nswer to the proble of rectivity, these odels rein usully difficult to ipleent. In ddition, the constructed odels hve only two sttes, stte of noinl opertion nd stte of filure nd do not coprise stte of degrded opertion. Mny works were relized to increse the nuber of sttes nd this by using Monte rlo siultions, but coputtion tie reins very long []. Fcing this fct, [33] introduces prognostic process bsed on the coupling of probbilistic representtion of the syste stte with n event representtion of the surveillnce of its coponents degrdtion. The process llows to predict the perfornce of the syste t instnt t + t, fro the observtion of the syste current stte t tie instnt t. The conception of the prognostic odel tkes plce in four steps: - Knowledge forliztion: this step consists of functionl nlysis nd dysfunctionl nlysis (AMDE nd HAZOP) in order to deterine two odels. The operting odel forlizes the opertion of the syste by using cusl nd qulittive reltions, nd reltions ong the different coponents. The dynic odel is bsed on the forliztion of the set of the coponents degrdtion processes of the syste by using Mrkovin processes. - onstruction of the probbilistic behviorl odel: this step consists of the integrtion of the operting odel nd of the dynic odel in set of unique forlis: the Byesin Dynic etworks (BD). This step is relized fro the generic echnis of construction. - onstruction of the eventul odel: This odel forlizes the knowledge of the syste current stte, of its coponents, s well s its different ctions of predicted intennce. - onstruction of the prognostic odel: the odel is constructed by coupling the two previous odels. Tht ens the integrtion of the eventul odel in the probbilistic odel, thus the result ppers s BD. The reliztion of prognostic on period of tie, begins by updting the eventul odel in function of the dt issued fro syste onitoring. The integrtion of the eventul odel in the prognostic odel llows to initilize nd to define siultion scenrio on tht period of tie. The siultion is then bsed on the teporl inference echnis nd on the 4
scenrio defined by the intennce opertions. This ethodology ws pplied on n experientl pressing syste [97]. The dvntge of this ethodology is tht it is pplicble to coplex syste nd not only to one of its coponents, nd prognostic is done in function of the intennce ctions. Hence, the prognostic odel constructed does not tke into considertion the odes of functioning to which the syste is subitted. I.8 - onclusion In this chpter coplete review of the prognostic pproches hs been presented. The dvntges nd disdvntges of ech of the three prognostic filies hve been lso detiled. They show the gret iportnce of these studies for the industril systes. The ethodology bsed on bci of degrdtion ws discussed nd showed, s consequence of this bibliogrphic study. For exple, the in proble of the experience-bsed pproch is tht it cnnot be pplied in the cse of new systes for which dt fro experience feedbck do not exist. Also, the pproches guided by dt lck rectivity when fcing chnge in usge conditions. When the pproches iss nlytic fors like those bsed on bci of degrdtion, they prove soe inflexibility during ppliction to vrious syste behviors. At the expense of cost, precision nd ccurcy re sought, thus the choice of novel physicl-bsed prognostic pproch, bsed on theticl odel of degrdtion, becoes n iportnt gol in prognostic. Therefore, precise, useful, nd elegnt theticl lws coe to our help in the following chpters in order to chieve the gol of this thesis. Our proposed odel is bsed on fous nlytic lws of degrdtion like Pris-Erdogn's lw which is lw of degrdtion by ftigue, nd Plgren-Miner's lw which is cuultive lw of dge. Despite this fct, long nd coplex nlyticl developent will be de in the following chpters to chieve novel degrdtion odel s tool for prognostic nlysis. Whenever such nlytic dge lws re vilble, the proposed pproch perits to deterine the Reining Useful Lifetie (RUL) of the syste. 43
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HAPTER II AALYTI LIEAR PROGOSTI MODEL OF DYAMI SYSTEMS 53
II. - Introduction Predicting the reining useful lifetie of industril systes becoes currently n iportnt i for industrilists knowing tht the filure which cn occur suddenly is generlly very expensive t the level of reprtion, of production interruption, nd is bd for reputtion. The clssicl strtegies of intennce [] re no ore efficient nd prcticl becuse they do not tke into considertion the instntneous evolving product stte, so it is iportnt to understnd the product in rel tie in order to prevent filure during opertion. In fct, we introduce prognostic pproch tht seeks to provide n intelligent intennce. In specilized literture, severl studies on prognostic procedure re presented, ong the we ention, the odel-bsed, sttistic-bsed, nd dt-bsed odels. The works bsed on bci of degrdtion s in the work of Peysson et l. [,3] re useful t this level. As the ltter is relted to the three influent coponents: process, ission, environent, it is nonnlytic bsed odel founded on expert knowledge nd on lrge dtbse. A proposed nlytic prognostic ethodology bsed on soe lws of dge in frcture echnics is developed here. The dges re generlly: crck propgtion, corrosions, chloride ttck, creep, excessive defortion nd deflection, nd dge ccuultion. Whenever their nlytic lws re vilble, the dvntge of prognostic pproch bsed on known dge lw for echnicl syste is tht it is dptble to new situtions nd useful in deterining the Reining Useful Lifetie (RUL) of the syste. The procedure proposed in this work belongs to the odel bsed prognosis pproch relted to the physicl odel. It is focused on developing nd ipleenting effective dignostic nd prognostic technologies with the bility to detect fults in the erly stges of degrdtion. Erly detection nd nlysis y led to better prediction nd end of life estitions by trcking nd odeling the degrdtion process. The ide is to use these estitions to ke ccurte nd precise prediction of the tie to filure of coponents. Erly detection lso helps void ctstrophic filures. Any prognostic ethodology ust lie on type of dge. In echnicl systes, the dge cn tke ny shpes. In this thesis, the cse of ftigue degrdtion hs been 54
chosen due to the fct tht it cn be theticlly forulted by vilble nlytic lws such s Pris-Erdogn's nd Plgren-Miner's lws. This pproch shows to be iportnt in ensuring high vilbility of industril systes, like in erospce, defense, petro-cheistry nd utoobiles. Aong these systes, the petrocheicl industries cn be cited s n exple of prognostic iportnce for the reson of fvorble econoic nd vilbility consequences on their exploittion cost [4]. Aong petrocheicl systes, pipelines serve to trnsport oil nd nturl gs between petrocheicl plnts. Their life prognostic is vitl in this industry since their vilbility is crucil here. The in cuse of filure for these systes is the ftigue due to internl pressure-depression vrition long operting tie. In pipelines study, the results of odel siultions re done for three cses of pipes: unburied, buried, nd offshore (under se wter). In utoobile industry, like for exple the suspension coponent, lso this pproch shows its iportnce for the se erlier resons s it is explined lter in this chpter. In vehicle suspension study the results of odel siultions re done for three cses of rod profile excittions. This chpter is orgnized s follows: first the echnicl odel of ftigue is presented in the liner cuultive dge cse then the prognostic odel of ftigue filure is developed nd finlly cse study of pipelines syste nd vehicle suspensions is illustrted. II. - Proposed Prognostic Model The purpose of this thesis is to construct process of prognostic cpble of predicting the degrdtion trjectories of coplex syste for given ission under given environent nd strting fro n initil known dge. The coplex syste is decoposble into sub-systes where ech one cn coprise dge function. The ftigue filure is one of the fous dge phenoen in echnicl systes like Figure. - Lod fluctution. 55
in ircrft where the wings re subject to the fluctution of ir pressure between xil vlue ( x ) nd inil vlue ( in ) (figure.) [5]. This type of lodings leds to crck propgtion tht cn ccelerte rpidly. Usully, icro-crcks exist originlly in the terils due to fbriction process where stresses rein fter nufcture. These icro-crcks re detected nd esured nd denoted by. The dvntge of the choice of ftigue dge for the developed prognostic ethodology is tht it is filure echnis very well studied in literture nd described under ny known nlytic lws. This echnis hs reltively the siplest forultion in coprison to the other dge phenoen. The ftigue chrcterizes the in filure cuse of industril equipents. II.. - Dge Evolution Lw The ftigue of terils under cyclic loding cretes icro-crcks. Strting fro n initil length corresponding to n initil cycle nuber, the cro-crcks becoe detectble nd unstble. These cro-crcks will grow under loding cycles to criticl length reched t cycles nd creting thus frctures tht led to filure. This evolution is represented in figure. in ters of norlized nuber of cycles / for siplicity of reding. () rck length Frcture Miniu detectble crck length Mcrocrck initition Figure. - Pre-rck ftigue dge. 56 It cn be ssued tht e / 8, where e nd ll re respectively the device diension in the crck direction nd the perpendiculr diension to the crck direction (figure.3).
is the crck length increent due to loding cycle d. t is the instnt corresponding to cycle nd to crck length. e rck length Syste stte Filure Filure level Filure Degrdtion t t t t + t t + t Tie t Figure.3 - rck length evolution. II.. - Pris-Erdogn's Lw The Pris-Erdogn's lw [6] perits to deterine the propgtion rte of crck length fter its detection. The lw of dge growth is given by eqution (): Where, d d ( K ) () nd re the teril nd environent preters. ( < << ); ( 4); is the crck length, is the nuber of cycles (where the RUL is derived directly), nd ΔK is the stress intensity fctor. It cn be distinguished (figure.4): - The long crcks tht obey to Pris-Erdogn's lw - The short crcks tht serve to decrese the speed of propgtion - The short physicl crcks tht serve to increse the speed of propgtion. 57
The lw cn be written lso s: log d d log + log( K ) () Phse I Low speed of propgtion Phse II Stble propgtion d / d ( K ) Phse III High speed of propgtion Figure.4 - The three phses of crck growth, Pris-Erdogn's lw. Fro the generl for of Pris-Erdogn's lw, McEvily A.J. nd Ritchie R.O. [7] hve proven the following for (eqution 3): Where eff x d d K K K, op ( K eff ) ( K ) (3) x K x : xiu stress intensity fctor, K op : stress intensity fctor required to open the ftigue crck. 58 So the decoupled for where two different functions of crck length nd of lod P cn be deduced: Where, d d The function: ( ) φ ( ) Y ( ) π φ( ) φ P nd the lod function φ P ) ( P) ; P K ; ( x ( ) (4)
with Y(): the geoetric fctor function of the body diensions, nd P : the lod preter. The Plgren-Miner's rule cn be used now to count the dges [8]. II..3 - Plgren-Miner's Rule The Plgren-Miner's rule [8] serves to copute the cuultive dge d i of different stresses levels i (i, i,..., ik) pplied for n i cycles. Knowing tht i is the totl cycle's nuber of stress i to be pplied, nd tht led to filure. The liner cuultive dge reltive to pplied stresses (i to k) is given by (5) (figure.5): D k k d k i i i ni i (5) D Filure uultive dge Relible n i / i Figure.5 - Plgren-Miner's liner rule of dge. II..4 - WÖhler's urve In teril ftigue, it is iportnt to know the criticl level of pplied stresses. When repeted stresses (t) re pplied long the tie under cyclic odel, they re liited between two extree vlues x nd in. WÖhler's curve governs the reltion between the pplied stress levels nd the criticl nuber of cycles during the ftigue process of the teril (figures.6 &.7). For exple, if the equipent is loded by stress level then the criticl cycle nuber is. Ech stress level hs its own criticl cycle nuber. 59
Applied stresses π sin t T x Men stresses Aplitude in ycle ycle t Figure.6 - yclic pplied stresses. The stress rnge: x - in The stress plitude: / The stress en: x + in Applied stresses : endurnce liit 3 3 Figure.7 - WÖhler's curve of ftigue. II..5 - Stress Intensity Fctor The stress intensity fctor is n iportnt ter in Pris' lw expression; it represents the effect of stress concentrtion in the presence of flt crck. When flt crck occurs in the syste body, the internl stresses in this section chnge fro unifor to non-unifor distribution round the crck (figure.8). This chnge is expressed by fctor K I clled the stress intensity fctor [9,] given, for ode-i crck opening (ode I: the crck opening is in the se direction of pplied stresses), by (6):. ( K ) Y ( ) I ( π ) ( ) φ ) φ ( P) (6) x ( 6
Figure.8 - on-unifor stress distribution ner crck. II..6 - Additivity Rule in Plgren-Miner's Rule. The cse where dge is cused by ftigue is n iportnt ppliction of the dditivity rule [,]. In this cse the esureent of dge is the length of the ftigue crck. The dditivity rule in Plgren-Miner's rule [8] hs been proposed s n epiricl rule in cse of dge due to ftigue controlled by crck propgtion. The rule sttes tht in ftigue test t constnt stress plitude i, dge could be considered to ccuulte linerly with the nuber of cycles. Accordingly, if t stress plitude the coponent hs cycles of life, which correspond to ount of dge, fter n cycles t stress plitude, the ount of dge will be ( n / ). After n stress cycles spent t stress plitude, chrcterized by totl life of cycles, the ount of dge will be ( n / ), etc. Filure occurs when, t certin plitude M, the su of prtil ounts of dge ttins the ount, i.e. when n n + n + + M M (7) is fulfilled. As result, the nlyticl expression of the Plgren-Miner's rule becoes: M i ni i (8) 6
Where i is the nuber of cycles needed to rech the specified ount of dge t constnt stress plitude i. The Plgren-Miner's rule is centrl to relibility clcultions yet no coents re de whether it is coptible with the dge developent lws chrcterizing the different stges of ftigue crck growth. The necessry nd sufficient condition for vlidity of the epiricl Plgren-Miner's rule is the possibility of fctorizing the rte of dge s function of the ount of ccuulted dge nd the stress or strin plitude p: d( ) d F( ) G( p) (9) The theoreticl derivtion of the Plgren-Miner's rule cn be found in Todinov []. A widely used ftigue crck growth odel is the Pris-Erdogn's power lw given by: Where, d( ) d ( K ) () K Y ( ) π : is the stress intensity fctor rnge; nd re teril constnts nd Y() is preter which cn be presented s function of the ount of dge. lerly, the Pris-Erdogn's ftigue crck growth lw cn be fctorized s in the previous stted eqution nd therefore it is coptible with the Plgren-Miner's rule. In the cses where this fctoriztion is ipossible, the Plgren-Miner's rule does not hold. Such s, for exple, the ftigue crck growth lw given by (): d( ) B γ d β D () discussed by Miller [], who chrcterises physiclly sll crcks. In the eqution bove: B nd β re teril constnts, γ is the pplied sher strin rnge, is the crck length t cycle, D is threshold vlue. 6
Thus, following wht hs been sid, the proposed odel cn use the dditivity chrcteristic of Pris' lw. II..7 - Mintennce nd Dignostic/Prognostic It is proved tht the schedule-bsed inspection/intennce DI (on Destructive Inspection) is less beneficil thn the on-dend (or continuous) inspection with pernently instlled sensors/condition bsed intennce SHM (Structurl Helth Monitoring) for ny resons like the incresed vilbility, quick ssessent of potentil/ctul dge events, incresing sfety, nd perfornce of dvnced terils. But the jor technicl chllenges for SHM reside in the sensors. The onitoring should be directed to the detection of the crcks nd corrosion, the ultiple dge odes, the pre-crck ftigue dge, nd the ccount for residul stresses. We cn sy tht the DI leds to prognostics bsed on the followings: - DI perfored t the tie of fbriction nd s in-service inspections - ondition bsed intennce-ctive coponent onitoring - Move fro dignosis to prediction of reining life nd structurl helth onitoring/ngeent. - Prognostics (for chinery) is the prediction of reining sfe or service life, bsed on n nlysis of the syste or teril condition, stressors nd degrdtion phenoen. For exple, bering crck fults y be prognosed by exining nd predicting their vibrtion signls. The reltion between intennce nd prognostic is surized by figure.9. Sensor dt Dignostic Module T t Prognostic Module (T?) BM Figure.9 - Dignostic-Prognostic-Mintennce BM: ondition-bsed Mintennce 63
II..7. - Flowchrt of Vrious oponents of Dignostic/Prognostic/Mintennce Process Inspection results Mteril properties estition oponent stte estition lcultion /Results Functions to be stisfied/ Liit sttes Estition of functions oponent evlution lcultion odel Dignostic Evolution odel Prognostic 64
II..7. - ycle of Prognostic-Dignostic-Mintennce Probbility of detection Mesured stte of structure urrent stte of structure Dge growth chrcteristics PROGOSTI Structurl Helth Monitoring Syste Filure Model Structurl Model DIAGOSTI Probbilistic prognosis of dge evolution (dge vs tie or cycles) Low Filure probbility within preset intervl High Inspection nd Repirs t intennce fcility 65
II..8 - Accuultion of Ftigue Dge In ftigue dge, to study the prognosis of degrded coponent, our ide is to predict nd estite the end of life of n equipent coponent subject to ftigue by trcking nd odeling the corresponding degrdtion function. To fcilitte the nlysis, it is convenient to dopt norlized dge esureent [,] D by using the dvntge of the cuultive dge lw of Plgren-Miner (figure.5). In fct, this lw helps estite the lifetie of coponents subject to lod cycles, it considers tht the dge frction stress level i is the rtio of n i over the totl cycle nuber i producing the filure. d i t For body of equipent of thickness e, tke the initil crck length s ( ). Knowing tht. e/ nd e/ 8, fro () recurrent for of crck length growth cn be deduced s [4]: d φ ( ) φ ( P ) j ; P j :is reliztion of P + φ ( ) φ ( P ); j where ( ) For the other sequences: + φ ( + φ ( ) φ ( P ) + φ ( ) φ ( P ) j j ) φ ( P ) j ( K ) d d I For ech cycle we hve: d, therefore: + ( K ) I As k k ni Dk di (Miner's lw with i in Miner's lw i i i in our odel) Bsed on the dditivity chrcteristic of Pris' lw, the ddition of dges gives the totl crck growth t filure point ( ) relized t the totl nuber of cycle : d At ech n i the crck grows of length Totl dge d d i, therefore the Miner's dge frction, for ny stress level (figure.), is given in ters of crck length by (): 66
67 ( ) d n d i i i Where, n i is the dge increent due to stress nuber i i is the totl dge for stress nuber i Then, the cuulted totl dge t cycle is given by (): () d d d D i i i i i i It cn be esily proved tht: ; ; ; ; ) : ( The other sequences re then : As : D D D D D D D D D D + d D i i Figure. - uultive stress levels. ()
68 A recurrent for of degrdtion cn be deduced s follows: ( ) ( ) ( ) ( ) ( ) ( ) (3) ) ( ) ( ) ( ) ( j j j j I I P D D Y Y Y K K D φ φ η π π π + + + + + + Where, ( ). ) ( ; ) ( ) ( ; ; j j P Y D D φ π φ η Hence, the new prognostic nlytic odel is presented by the generl function given by (4): And therefore, the degrdtion trjectories D() long the totl nuber of loding cycles cn be drwn [3]. ( ) (4) ) ( ) ( ) ( j rog Y P D D π +
II..9 - Flowchrt of the Prognostic Model The following flowchrt surizes ll the procedures of the proposed odel [4]: Dignostic/ Inspection Input initil preters: (, e, diensions,, ) Estition e/8 For ech lod cycle, Anlytic siultion of crck growth Lod siultion: j lculte crck length by recurrence: + φ ) φ ( ) ( j lculte reining useful lifetie t cycle : RUL() - Degrdtion ccuultion lculte degrdtion vlue t cycle : D D < Yes o Plot ( D ; ), ( ) Plot ( RUL( ) ; ), ( ) Prognostic 69
II.. - Environent Effects in the Proposed Prognostic Model The environent effects re tken into ccount through two preters nd. These preters re relted to the teril in its environent. Lrge vlues of (>4) correspond to the cse of brittle terils (brittle filure), nd sll vlues of ( ) correspond to the cse of ductile terils ( fully plstic). Otherwise for ftigue filure the rnge vlue of is: 3. The preter depends inly on the specien length. For lower toughness steels is greter thn or equl to 3 [5]. oefficient is ffected by the edges nd consequently its vlue depends on whether it is the cse of plne stress or plne strin. However, for the cse of n infinite equipent body nd fr fro the edge effects, the coefficient tkes constnt vlue [6]. nd depend on the testing conditions, such s loding rtio in / x, on the geoetry nd size of the specien, nd on the initil crck length. These two preters govern the behvior of the teril during the ftigue process through the crck propgtion. The environent influencing preters on this process like teperture, huidity, geoetry diensions, teril nture, wter ction, soil ction, pplied lod loction, body shpe, re lso represented by these two preters nd. These two preters re evluted by the en of experients in true conditions. Exples [5,6]: 5.. -3 (free ir).3. -4 (under soil). - (offshore) nd 3 (etl). 7
II.3 - Appliction of the Prognostic Method to Industril Systes To illustrte the proposed new nlytic pproch, it will be pplied in this section to two iportnt echnicl systes which re: the pipelines syste in petrocheicl industry nd the suspensions in utootive industry. The prognostic studies of these two fields of industry re essentil for econoy resons. II.3. - Vehicle Suspension Ftigue Life Ftigue nlysis of vehicle suspension (figure.) by finite eleents odels ws done in ny works [7] beside the experientl results. It perits to define the loction of potentil ftigue crcks. The Plunge type Rubber ounting jor feture of locl strin c.v. joint ftigue lives to crck initition. The originl c.v. joint theories were developed for unixil stress conditions, nd lter, to eliinte the errors due to the siplified unixil conditions. It ws proposed in literture [8,9] tht for high cycle ftigue successful life estites for bixil stress conditions could be de using cobintions of xil nd sher stresses. Finl-drive housing - bolted to sub-fre Eccentric djuster for toe-in Figure. - Vehicle suspension syste. Sub-fre There is uch experientl evidence fro ftigue testing crried out in the iddle of the lst century showing tht stress grdients hs n iportnt effect on the totl ftigue life of coponent. Stress grdients hve lso been used in n ttept to explin the effect of notch sensitivity. Endurnce liit, noinl stress Se Blunt notch S e e /Kt S e e /Kt Shrp notch S e S th Figure. - Reltionship between endurnce liit stress e nd the stress concentrtion fctor Kt [3]. Kt 7
Finite eleent nlysis provides surfce strins on the odel but for rel engineering coponents it is very difficult to deterine the stress concentrtion fctor t notch (figure.). The stress concentrtion fctor is the se s the stress intensity fctor explined in prgrph II..5. The endurnce liit stress is the stress level for which the criticl nuber of loding cycles tends to infinity (refer to prgrph II..4). Where, e is the sooth specien endurnce liit stress, S th is the threshold stress for nonpropgtion crcks, i.e. below S th ftigue is not influent nd S e S th Kt is the stress concentrtion fctor, Endurnce liit, noinl stress S e S e e /Kt Totl life rck initition Figure.3 - Reltionship between endurnce liit stress e nd the stress concentrtion fctor Kt for crck initition nd totl life. Endurnce liit of notch free specien nd we hve: Kt. Endurnce liit of notched specien The endurnce liits [9] re obtined fro stndrd rotting be experients crried out under certin specific conditions. It is given by: S e e / Kt. As the stress concentrtion fctor increses, nd tht for ny ductile etls, iniu vlue of ftigue liit stress occurs nd is S th. Further incresing the stress concentrtion fctor by shrpening the notch produces no further reduction in ftigue strength (figure.3). 7
The prts foring the vehicle suspension re indicted in figure.4 where the dper's eleent cn be seen. rck loction in suspension Figure.4 - Vehicle suspension coponents nd crck possible loction Using test dt on plte nd round br speciens in luinu lloy nd steel terils hve shown tht if ftigue life to first crck initition is considered, then the ftigue strength reduces with incresing stress concentrtion with no liiting vlue (figure.5). Mny works [,,] hve shown tht the constnt plitude endurnce liit does not pply to the nlysis of rel service loding if soe cycles in the loding exceed the constnt plitude endurnce liit stress plitude. For finite life design the lrger cycles in the loding cuse the endurnce liit stress to be reduced significntly, with the result tht sll cycles contribute to the ftigue dge process. Figure.5 below [] shows the results of strin-controlled constnt plitude tests on n luinu lloy t high teperture. The Finite Eleent clcultion de by the softwre SAFE (FE-SAFE) fro n elstic Finite Eleent Anlysis (FEA) shows excellent correltion for high cycle ftigue. For low cycle ftigue, t cycles the clculted ftigue life is conservtive by fctor of 3. This is coonly observed phenoenon t such low ftigue lives in coponents where yielding occurs cross the entire section. For coprison, n elstic-plstic FEA nlysis of the odel ws used s input into the FE-SAFE nlysis, nd the correltion with the test result ws then excellent. 73
ycles Figure.5 - oprison of test dt with clculted lives fro elstic nd elstic-plstic FE nlysis. This coponent ws nlyzed in FE-SAFE nd copred with the results of ftigue testing. A scle fctor ws pplied to the test loding to produce filure. The correltion between the clculted life of 63 repets of the lod history nd the test life of 65 repets is extreely good. The steel coponent ws nlyzed [] with lod-tie history in one direction (figure.6). A scle fctor ws pplied to produce filure. The nlysis used stresses fro n elstic FEA; ftigue lives were clculted for ech node on the odel, using verged nodl stresses. Experience hs shown tht this is uch ore ccurte thn using stresses t integrtion points or t the eleent centroid. Loding Sples 74 Figure.6 - Loding history for ccelerted testing (left) nd ftigue life contours (right). Test life: 65 repets of loding. lculted life: 63 repets of loding.
In designing engine crnk shfts (figure.6), the finite eleents nlysis is used to generte stress solutions. The FEA nlysis shows tht the principl stresses chnge their orienttion nd gnitude during the lod cycle pplied to the crnk shft. FE-SAFE uses the sequence of FEA nlysis results to clculte the ftigue life t ech node. FE-SAFE correctly identified the criticl loction in the crnk shft, using Brown-Miller ftigue nlysis, nd correlted well with test results. A coon thee fro these vlidtion exercises is tht unixil strin-life using the xiu principl stress cn fil to identify the criticl loction, for coponents where bixil stresses (Von-Mises) nd prticulrly non-proportionl stresses re present t the criticl loctions. In the coputer-bsed ftigue nlysis of the finite eleent odel the type of loding depends very uch on the custoer's requireents. Soe copnies [] specify vlidtion using siple sinusoidl loding, wheres other copnies, such Ford, require the ppliction of esured tie histories of verticl, brking nd cornering forces on the tyre contct ptch or wheel center (figure.7). At present, the test procedure uses single ctutor to pply the forces t the tyre contct ptch, ngled to produce specific reltionship between the three forces. FE-SAFE llows for different tie histories to be pplied in ech direction, up to 496 lod histories of unliited length. Figure.7 - Appliction of force tie histories. 75
II.3.. - Types of Mechnicl Effects, Their Mechniss, nd Possible onsequences The following flowchrt describes the reltionship between the sources, the echnicl effects nd the consequences of vrious loding stresses [5]. Stressors Ageing echniss onsequences Stress onstnt reep Degrdtion (Dge) Strin onstnt Stress vrible Relxtion Ftigue Defortion Teperture Irrdition Therl Ageing Irrdition Dge Ebrittleent nd rcking orrosive ediu orrosion Mteril Loss Reltive Motion of Fluids nd Solids Wer & Erosion II.3.. - Autotic Dignostic of Bd Suspension Bushing Autoobile suspension bushings coe in vriety of shpes, sizes nd thicknesses, ccording to their ppliction. Bushings y be de fro severl terils, including rubber, polyurethne, urethne nd grphite coposites. Bushings prevent wer to expensive suspension coponents by bsorbing verticl nd lterl forces produced by the vehicle over different terrin. They cushion nd bsorb shock on the chssis to keep it shock fro entering the pssenger coprtent. While bsorbing these vibrtions, they still llow liited oveent nd flex in the suspension joints, keeping the wheels firly grounded nd on trck during turning noeuvres. A vehicle's owner y check ll its suspension bushings for proper shpe nd condition. 76
II.3..3 - Prognostic Study for Vehicle Suspension Systes Let us consider hlf-vehicle suspension syste (figure.8) subject to non-regulr rod surfce excittions [3]. It is coposed fro front prt nd rer prt. To study the prognostic of this syste, it is iportnt to define the echnicl odel in order to conclude the output response fro the input excittion rod. enter of ss K b c b r body K c Rer wheel Front wheel re given by: The dynic equtions of the syste Rod irregulrity x + (f c + f k ) + (f cb + f kb ) I θ + l (f c + f k ) l b (f cb + f kb ) (f c + f k ) + k ( x w ) x (f cb + f kb ) + k b ( x b wb ) b x b x ( l b x l l + l b + l x b )/ l, ( x tn θ θ x l b ) x b l b θ l x l x f ci c i ( x ), i, b f ki k i ( xi i ), i, b x i xi Figure.8 - Vehicle suspension odel Where, : vehicle ss, I : oent of inerti : ss of front wheel, b : ss of rer wheel θ : rotry ngle of vehicle, x : verticl displceent i : c friction coefficient of duping ( i, b ) f c, f cb : duping force of the front/rer wheel f k, f kb : restoring force of the front/rer wheel 77
k, k b : spring constnts of the front/rer suspension k, k b : spring constnts of the front/rer wheel x x b x xb, : verticl displceent of the front/rer wheel, : displceent of the vehicle body t front/rer wheel l, l b : distnce of the front/rer suspension to center w, w b : irregulr excittions fro the rod surfce (See figure.9) x ( w ) b b x ( w ) Figure.9 - Rod profile excittion. II.3..4 - Syste Identifiction The odel preters re given by the following nuericl dt [3]: kg, I kg. 3 kg, b 5 kg c b 4 //s, c 5 //s k 56 /, k b 4 / k k b 5 k/, l.9, l. b The trix for of the previous equtions is given by (5): M z + z + Kz Eu (5) Where M is the ss trix, is the duping coefficients trix, nd K is the stiffness trix. 78
79 The input excittion vector is: [ ] T w w b u The output dper displceent vector is: The verticl ccelertions b b x x x x,,, re esured vribles. The trices K M,,, nd E re given by: b b l I l I l l l l M / / / / b b b b b b b b c c c c c l c l c l c l c c c c + + b b b b b b b b b k k k k k k k l k l k l k l k k k k K b k k E The stte vectors (dper displceents nd velocity) re: ) ( ) ( t z t z x, ) ( ) ( t z t z x II.3..5 - Ftigue Dge Modeling of Suspension The odeling of the suspension dge begins by deterining the stress intensity fctor coposed of the ultipliction of two functions: Where, ) ( φ is the crck length function deterined in ters of geoetric function Y(), ) ( j P φ is the loding function. [ ] T b b x x x x z ( ) ( ) ( ) ( ) (6) ) ( ) ( j x I P Y K φ φ π
Assue tht the front suspension of the syste hs crck length perpendiculr to the exterior lod (figure.). f k, f c x x ll e f k, f c Figure. - Suspension ftigue crck odeling. Let be the teril constnt, then: ( Y ( ) π ) φ ( ) by [4] (6): Therefore, by epiricl esureents, the first function cn be considered s given φ ( ) ( π )..4 e + 7.33 e 3.8 e 3 + 4 e 4 (6) Where, : the crck length t cycle, e : the width of the echnicl coponent of the suspension. e Assue tht the xiu of is: ; [5] 8 8 We define D (s << ) e ed We replce φ in eqution (3) nd we get: 8 D D + η φ( D j ) φ ( P ) (7) 8
Knowing tht Pj is the lod preter, nd we hve φ ( P j ) Pj Pj. Moreover, η is 6 teril constnt nd we hve η 8. [4]. II.3..6 - Siultion of the Degrdtion Model We will siulte the degrdtion odel by generting the lod P j of rod profile [ w ] T w b [3] under the Gussin orl lw for the three odes of rods (tble.). Fro the syste of equtions (5), the solution of this syste of trices gives the output vector z. Then, the rnge of the suspension displceent is given, for the front wheel, by (8): x j x j x j (8) We tke s en vlue x nd stndrd vrition j xj, we obtin set of { x } r for ech rod ode ( r,,3 ), the lod preter is lwys P j. We hve the recursive forul (9) in ters of crck length: ( j + φ ) φ ( P ) (9) With [5]: ; η ( ) ; η 8. 6 φ ( nd )..4 e 3 ( π ) φ ( P ) P P. j j j j + 7.33 e 3.8 e + 4 e 4 () (): Where, The plitude of the stresses developed in the suspension due to x j is siplified by j E x : the length of the suspension device ( 5 ) x j : the vrition of this length (diltion) under rod profile excittion E : Young's odulus of the suspension teril ( E GP) j () 8
given by: Therefore, the recursive expression of the crck length for the suspension odel is 3 4 + 4 j ( π )..4 + 7.33 3.8 + e e e e () Fro the eqution D, the recursive expression of the degrdtion indictor for the suspension odel becoes: D( ) ( π )..4 7.33 3.8 + 4 + (3) e + e e 3 e 4 j II.3..7 - Siultion of Three Rod Profiles To tke into ccount vrious stte of rods, we consider three different types of rods which re: severe, fir, nd good. In the following tble, we indicte the sttisticl chrcteristics of ech type of rods. Tble. - Sttisticl chrcteristics of ech ode of rods Rod Mode Severe (ode ) Fir (ode ) Good (ode 3) Men of x oefficient of Stndrd Lw j Vrition of Devition ( x j in ) x j in % (in ) 5% 5 orl 5 % 5 orl 5 5%.5 orl The prbolic rod profile for T seconds of vehicle circultion tie s recurrent intervl is considered. And this intervl is repeted s needed until reching the filure (D ). Figure. illustrtes the rod profile: 8
Rod profile x j x j Tie t T (s) T (s) T (s) Figure. - Siulted rod profile Ech intervl shows tht the rod profile contins syetric curve of width T/8.5(s) with pek vlue followed by horizontl run of zero plitude. II.3..8 - Siultion Results The prognostic study of suspension is relized through the degrdtion siultion (eqution 3). The ethodology is coposed of two prts: In the first prt, the siultion of the rod profile for the three odes (severe, fir, nd good) (tble.) is done using the orl lw fro which x nd re deduced. In the second prt, the crck length is cuulted t ech cycle (eqution ). The resulting curves D() re represented in the following three figures: 83
Figure. - Degrdtion trjectory for the rod with ode profile. In ode cse (Severe), it is noted tht (figure.) for 6,836, cycles, the degrdtion D reches the criticl vlue D. The deduced lifetie of the suspension is 6,836, cycles of rod excittion in ode. Moreover, the first sign of dge ppers t bout,5, cycles. Strting fro 6,, cycles, the slope of the degrdtion curve becoes very cute; hence dge is incresing very fst. Figure.3 - Degrdtion trjectory for the rod with ode profile. In ode cse (Fir), it is noted tht (figure.3) for,85, cycles, the degrdtion D reches the criticl vlue D. The deduced lifetie of the suspension is,85, cycles of rod excittion in ode. Moreover, the first sign of dge ppers t 84
bout 4,, cycles. Strting fro,, cycles, the slope of the degrdtion curve becoes very cute; hence dge is incresing very fst. Figure.4 - Degrdtion trjectory for the rod with ode 3 profile. In ode 3 cse (Good), it is noted tht (figure.4) for 7,, cycles, the degrdtion D reches the criticl vlue D. The deduced lifetie of the suspension is 7,, cycles of rod excittion in ode 3. Moreover, the first sign of dge ppers t bout 6,, cycles. Strting fro 6,, cycles, the slope of the degrdtion curve becoes very cute; hence dge is incresing very fst. In ddition, figure.5 recpitultes the three previous figures. Figure.5 - Degrdtion trjectory for the three odes of rods profiles. 85
II.3..9 - Anlysis of the Siultion Results The expecttion of the lifetie for ode is nerly 63% of tht of ode nd the expecttion of the lifetie for ode is nerly 63% of ode 3 (figure.5). It cn be noticed fro the obtined results tht the increse of the suspension lifetie reltive to the rod of ode 3 is s follows: ode ()/ode (3) 5 % nd ode ()/ode (3) 59 %. Fro the bove, the three expected lifeties re s follows: 6,836, cycles;,85, cycles; 3 7,, cycles. Then, our prognostic procedure yields the Reining Useful Lifeties (RUL) for the three odes (figure.6) tht cn now be esily deduced fro these three curves t ny instnt or ny ctive cycle s follows: For ode : RUL () - ; For ode : RUL () - ; For ode 3: RUL 3 () 3 - ; 8 x 6 6 Reining Useful Lifetie RUL (cycles) 4 8 6 4 Mode 3 : Good Mode : Fir Mode : Severe..4.6.8..4 Degrdtion D Figure.6 - Reining Useful Lifeties estited by the prognostic odel. 86
II.3.. - onversion of RUL into Yers To convert the suspension lifetie into yers' unit, knowing tht ech cycle durtion is seconds (refer to figure.), then: RUL(s) RUL(). We ssue tht the suspension tie usge is % of dy, which corresponds to.4 hours/dy. The conversions fro ycles to K nd to Yers, for vehicle running with 5 k per hour, re given by the following literl expressions: Fro ycles to K: RUL(ycles) (s/ycle) 5(K/hour) RUL(K) 6(s/in) 6(in/hour) Fro K to Yers: RUL(ycles) 36 (ycles/k) RUL(K) RUL(Yers).4(hours/dy) 5 (k/hour) 365(dys/Yer) RUL(K) 43,8 (K/Yer) Therefore, the RUL results cn be expressed by the following units: ycles, or K, or Yers. Thus, the expected lifeties' durtions re: 6,836,(cycles) (s) For ode : 4.34 yers 6(s) 6(in).4(hours) 365(dys) For ode :,85,(cycles) (s) 6.88 yers 6(s) 6(in).4(hours) 365(dys) For ode 3 : 7,,(cycles) (s).9 yers 6(s) 6(in).4(hours) 365(dys) Moreover, the vlidtion of these results cn be found in the work of reference [6] on the ftigue life of suspensions. An verge life of, k is deduced under severe conditions nd which corresponds to 4.57 yers for vehicle running with 5 k per hour nd for.4 hours per dy. 87
II.3. - Prognostic Study for Pipelines Systes II.3.. - Introduction Pipelines re petrocheicl systes tht serve to trnsport oil nd nturl gs between sites. Pipelines tubes re considered s principl coponent in petrocheicl industries, their life prognostic is vitl in this industry since their vilbility hs crucil consequences on the exploittion cost. The in filure cuse for these systes is the ftigue due to internl pressure-depression vrition long the tie. These pipelines re usully designed for ultite liits sttes (resistnce). Moreover, buried pipelines re subject to corrosion due to soil ggression effects. They re nufctured s cylindricl tubes of rdius R nd of thickness e. Figure.7 - Buried pipes. The DV rules propose for pipelines trget probbility of filure bout -5. Their in filures re due to seisic ground wves, soil settleents, buckling, defortions, internl nd externl corrosion, stress concentrtion in welding nd fitting, vibrtion nd resonnce, pressure fluctution over long period. The ftigue filures by crcks propgtion re detected by crcks detection tools. A significnt prt of in pipelines re subjected to externl crcking, which is serious proble for the pipeline industry like, for exple, in Russi [7], U.S., nd nd [8]. Identifiction of externl crcks is chieved using different ondestructive Evlution (DE) ethods. If crcks re reveled during inspection, their influence on the reining useful lifetie (RUL) of the pipeline should be ssessed in order to choose wht intennce ction should be used: do nothing/repir/replce. Pipeline integrity is ssessed on the ssuption tht soe defects fter In-Line Inspection (ILI) y be: still undetected; detected, but not esured; detected nd esured. Three cse studies of pipes re considered here: unburied, buried (figure.7) nd subse (offshore pipes). Ech one of these situtions requires different physicl preters like: corrosion, soil pressure nd friction, wter nd tospheric pressure. 88
II.3.. - Pipes Stress Modeling e/r /. The pipes re cylindricl thin tubes since their thickness e to rdius rtio is [9]: θ L θ R L e L θ Pipe cross-section Figure.8 - ylindricl pipelines. In this cse, the stresses due to internl pressure P re of ebrnes types without ny bending forces. The stresses re circuferentil (hoop stress) θ nd longitudinl (xil stress) L (figure.8). They re given by (4): θ P R ; L e P R e (4) The criticl position of crcks is longitudinl which is perpendiculr to the direction of xil stresses θ. The crck hs depth (or length) esured in the thickness direction (figure.9). Generlly, the following rtio intervl cn be considered:. /e.99 R θ e e - θ Figure.9 - rcked pipe section..3): We cn illustrte ll stresses types in pipe body by the following figure (figure 89
Body internl crck Internl rdius Thickness Body externl crck Ebedded crck Figure.3 - Stresses types distribution in pipe body It is entioned here tht only the first ode of crck (K K I ) is considered, i.e. the opening ode (the other odes re sliding nd tering ode). II.3..3 - Stte of Stresses in the Tube Body The tubes re odeled s cylindricl shells of revolution. When thin tubes of rdius R nd of thickness e re under internl pressure P, the stte of stresses is ebrne-like without bending lods. The ebrne stresses re circuferentil (hoop stress) θ nd longitudinl stresses (xil stress) L (figure.3).these stresses re given by (4). Figure.3 - Axil nd hoop stresses in pipes. The criticl crcks re those which re perpendiculr to xil stresses θ (figure.3), tht ens longitudinl crcks which re prllel to the tube xis. A crck is of depth or of length, esured in the direction of the tube thickness e R - R. R is the externl rdius nd R is the internl rdius of the pipe. 9
θ θ Figure.3 - rck length in rdil direction. II.3..4 - Stress Intensity Fctor The stress intensity fctor for tubes is given by []: K I φ ( ) φ ( P) Where : φ ( ) K K I I K.6 I ( Y ( ) π ) ; φ ( P) ( ) ( PR / e) ; Y ( ) + ( / e) 3 ( / e) π ( PR / e) θ.6 + ( / e) 3 ( / e) Where, Y() is the geoetric fctor function of the pipeline geoetric preters (, e), : is the tencity of teril (or criticl stress intensity fctor) nd is given by: K I (5) (6) K I J I E ( ν ) (7) Where, J I is the resistnt crck force of the teril; E is the Young's Modulus ndν is the Poisson rtio. ote tht the fctor K I ust not exceed the vlue of K I [4], nd 3. II.3..5 - Degrdtion Model Expression of Pipes With, Fro the stress intensity fctor defined bove: + φ ) φ ( p ) ( j ; where p j is the siulted lod. 9
φ ( ) ( ) ( / ) ( ) + e Y π.6 3 ( ) / e ( ) ( P R ) 3 p j ) p j j j / φ ( e π 3 Then the dge ccuultion is given in ters of the crck length by the following recursive reltion: +.6 + ( / e) 3 ( / e) π j e 3 3 ( P R / ) (8) And the degrdtion indictor of the pipe cn be written s in (9): D D + η φ( D ) φ ( p j ) ; where η ( ) D( ) ( π ).6 + ( / e) 3 ( / e) 3/ 3 + ( Pj R / e) 3 (9) II.3..6 - Siultions of Three Levels of Internl Pressure onsider pipe of rdius R 4 nd of thickness e 8 trnsporting nturl gs. In this cse, the preters re: 5. -3 (free ir),.3-4 (under soil), - (offshore), nd 3 (etl). Tke the initil crck length.. The internl pressure P j is siulted following tringulr for to be siilr to the rel cse of pipelines operting condition (pressure-depression) (figure.33). For ll three pipes cses, the function φ ( ) nd the preter re the se [4]. Internl Pressure P j P P t T T T Figure.33 - Tringulr siultion of internl pressure Three xil levels of P j re considered which re P 3 MP, 5 MP, nd 8 MP nd with repetition period T. At ech of these levels, degrdtion trjectory D() is 9
deduced in ters of the cycle nuber. When D reches the unit vlue, then the corresponding is the lifetie of the pipe in ftigue cse. For siultion purposes, in tble., the en vlues of pressure P j re considered s the xil vlues P. The coefficients of vrition re δ Pj. Tble. - Sttisticl chrcteristics of ech pressure ode. Pressure Mode P j (MP) δ Pj (%) Lw High (ode ) 8 % Tringulr Middle (ode ) 5 % Tringulr Low (ode 3) 3 % Tringulr The siultion of the nlytic prognostic odel (eqution 9) is executed for ech level of internl pressure (high, iddle, nd low). The estition of rel lifetie syste necessittes huge ount of pressure siultions of order of hundreds of illions; hence, n pproxited odel of lifetie siultion of order of,, itertions hs been used. onsequently, high cpcity coputer (ORE i7, 3 GHZ icroprocessor with n 8 GB RAM) hs been considered for this purpose. Usully, the pipelines y be plced in prctice in three dispositions: onshore (unburied, buried), nd offshore (under wter) [3]. II.3..7 - Unburied Pipe se This sitution [3] is suitble outside cities between sttes nd countries where they do not intercept ny construction or trnsporttion fcilities. In this cse, the norl service lod includes only the internl pressure. The results of degrdtion trjectory siultion (9) re shown in figure.34 below. The pipe lifeties for this cse re nerly 3.3 yers for ode (high pressure), 4.68 yers for ode (iddle pressure), nd 6.85 yers for ode 3 (low pressure). In coprison with previous lifeties' 93
studies on pipelines [3], it cn be concluded tht in reltion with pipes diensions, internl pressure, nd pressure cycle, the order of gnitudes of the present vlues re relistic. It is noted tht t the beginning (between yer nd yer) ll odes give the se degrdtion level of.5 where crck lengths re negligible when copred with the criticl crck length. Figure.34 - Degrdtion evolution of unburied pipes under three odes of pressure. For three odes of internl pressure, the Reining Useful Lifeties for the unburied tubes re evluted in yers nd illustrted in the figure.35. It is noted tht these three curves re decresing fro their corresponding globl lifetie to zero vlue where the degrdtion reches the unit vlue D. 94
Figure.35 - RULs evolution of unburied pipes under three odes of pressure. II.3..8 - Buried Pipe se This cse is useful for ny resons (reduce plnt congestion, fewer pipe bending, protection fro bient teperture chnges, wind nd other lods) [33]. This study is liited here to norl service lods tht include only internl pressure P int nd soil ction (figure.36). Figure.36 - Forces on buried pipe under soil. The soil effects on the pipe surfce re [33]: the norl force S nd the soil friction F given by: 95
S P A ext p S da 4 Rγ S π R F µ S L, F ( H R) F π R + W P (3) (3) (3) P int P ext da: differentil contct re. θ opute the xil stress: L θ + 4.4P ( P eq int P e ext + < L ) R Pint R ; L L e θ ext L < θ θ for P θ, F int, in (33) 3 MP x Figure.37 - Internl nd externl pressure in buried pipes. θ The effects of the force S on the pipe surfce is expressed by n externl pressure Pext tht opposes the effects of n internl pressure P int. Siilrly, the effects of the friction force F on the pipe surfce ( L,F ) oppose the effects of the internl pressure P int ( L ) (figure.37). The depth of the pipe is tken H 7R nd the friction coefficient intervl is [4]:.5 µ.7. The soil specific weight is γ 9.843 kg/c. The weight per liner eter of pipe nd gs content is given by eqution (34): W p π R eγ pipe + π R γ gs 3.7 kg/. (34) The specific grvity of the pipe teril nd of the nturl gs re respectively: γ pipe 7,85 kg/ 3 nd γ gs 6 kg/ 3. Fro the siultion of the proposed nlytic prognostic odel (9), the pipe lifeties re deduced fro figure.38. They re 8.33 yers for ode (high pressure),.87 yers for ode (iddle pressure), nd 7.35 yers for ode 3 (low pressure). It is noted tht t the beginning (between yer nd 3 yers) ll odes give the se degrdtion level of.5 where crck lengths re negligible when copred with the criticl crck length. Previous 96
pipes lifetie studies [34] show tht in reltion to the pipes diensions, pressure levels nd pressure cycles, the order of gnitudes of these obtined vlues re relistic. Figure.38 - Buried pipe degrdtion function of lifetie for the three odes of internl pressure. The Reining Useful Lifeties in yers re lso evluted for buried tubes for three odes of internl pressure nd they re illustrted by the figure.39. We note tht these three curves re decresing fro their corresponding globl lifetie to zero vlue where the degrdtion reches the unit vlue D. Figure.39 - Buried pipe RULs function of degrdtion for the three odes of internl pressure. 97
II.3..9 - Offshore Pipe se In this cse, the sitution where the pipes re under se wter (offshore pipeline) serving to trnsport oil or gs fro rine offshore to refinery plnt is considered [35,36,37]. They re subject, beside internl gs pressure, to externl wter nd tospheric pressure (figures.4 &.4). Figure.4 - Offshore pipelines network. Figure.4 - Offshore types for vrious depths. 98
onsider pipe (figure.4) of dieter φ 48 nd of thickness e 8, the externl pressure round the offshore pipe is given by (35): P ext P P ext W + P t ρ g H + t w 6,63,95 P 6.6395 MP (35) Where, The depth of offshore pipe considered here is: H 6. Atosphere pressure t se level t.35 MP. The specific weight of sewter is: ρ w,3 kg/ 3. The grvittionl ttrction is: g 9.8 /s. Then, the net xil stresses in the pipe body re given by (36): θ ( P int Pext ) R e (36) Figure.4 - Offshore pipe preters. After the siultion of the proposed prognostic odel for the offshore pipeline degrdtion nd under three levels of internl pressure P int, the pipe lifeties re illustrted by figure.43. They re.7 yers for ode (high pressure), 4.84 yers for ode (iddle pressure), nd.69 yers for ode 3 (low pressure). It is noted tht t the 99
beginning (between nd 5 yers) ll odes give the se degrdtion level of.5 where crck lengths re negligible when copred with the criticl crck length. Se rerk, like in the previous two cses, pplies for the relis of these lifeties results [3,34]. Figure.43 - Offshore pipe degrdtion function of lifetie for the three odes of internl pressure. The Reining Useful Lifeties re evluted in yers for offshore tubes under three odes of internl pressure nd we deduce figure.44. We note tht the RULs curves re decresing fro their corresponding globl lifetie to zero vlue where the degrdtion reches the unit vlue D. Figure.44 - Offshore pipe RULs function of degrdtion for the three odes of internl pressure.
II.4 - onclusion An nlytic prognostic odel is introduced in this chpter tht perits to predict the Reining Useful Lifetie (RUL) of dynic systes. This odel considers the ftigue s dge preter nd hence it is bsed on well known lws of dge like Pris' nd Miner's lws. An index of degrdtion ws derived tht vries fro zero to one. Our proposed odel is bsed on the link between this index D nd the crck length. Filure is produced when reches criticl length. Hence, our odel is given by siple function relting the instntneous degrdtion to ctul crck length s esureent of ctul dge. Our i is to evlute the evolution of the syste lifetie t ech instnt. For this purpose the degrdtion trjectories hve been used in ters of cycle nubers or the tie of opertion. Fro these degrdtion trjectories, the RULs vritions re deduced. The prognostic of coplex syste cn be deduced fro the prognostic of its sub-systes when their dge lws re vilble. To deonstrte the effectiveness of our odel, two industril exples hve been considered in siultion in this chpter. These systes re the vehicle suspension systes nd the petrocheicl pipelines. For the vehicle suspension, three odes of rod profiles re siulted. For the pipes, three types of pipes hve been considered: unburied, buried, nd offshore, nd three odes of internl pressure re exined. In such industril systes, this odel proves tht it is very convenient nd it provides useful tool for prognostic nlysis. Moreover, it is less expensive thn other odels tht need lrge nuber of dt nd esureents. In the following chpters we will enlrge this study by considering the nonliner cse of cuulted dge nd the probbilistic influence of the bsic preters on degrdtion nd on RULs evolution.
References [] G. VAHTSEVAOS, F. LEWIS, M. ROEMER, A. HESS, nd B. WU, Intelligent Fult Dignosis nd Prognosis for Engineering Systes, John Wiley & Sons, Inc., 6, ch. 5,6, nd 7. [] F. PEYSSO, M. OULADSIE, R. OUTBIB, J-B. LEGER, O. MYX, nd. ALLEMAD", A generic prognostic ethodology using dge trjectory odels", IEEE trnsctions on relibility, vol. 58 (no. ), June 9. [3] D. HELIDZE nd J.P. USUMAO, "A dynicl systes pproch to filure prognosis", J. of Vibr. nd Acoustics, Vol. 6, pp. -8, 4. [4] K. EL-TAWIL, A. ABOU JAOUDE, S. KADRY, H. OURA, nd M. OULADSIE, "Prognostic bsed on nlytic lws pplied to petrocheicl pipelines", Interntionl onference on oputer-ided Mnufcturing nd Design (MD ), hin, oveber. [5] J. LEMAITRE nd J. HABOHE, Mechnics of Solid Mterils. ew York: bridge University Press, 99. [6] P. PARIS nd F. ERDOGA, "A criticl nlysis of crck propgtion lws," Journl of bsic engineering, Trnsctions of the Aericn society of echnicl engineers, Vol. 85, o. 4, pp. 58-534, 963. [7] A.J. MEVILY nd R.O. RITHIE, Ftigue Frct. Eng. Mter. Struct. (988) 847-855. [8] M. A. MIER, uultive dge in ftigue, Journl of Applied Mechnics, vol., A59-A64, 945. [9] M. LAGO, Introduction l Ftigue et Mécnique de l Rupture, entre d essis éronutique de Toulouse, ESIA April, 999. [] J. LEMAITRE nd R. DESMORAT, Engineering Dge Mechnics, ew York: Springer-Verlg, ch. 6, 5. [] M. TODIOV, ecessry nd sufficient condition for dditivity in the sense of Plgren-Miner rule, oput. Mter. Sci., vol, no, pp -,. [] M. T. TODIOV, Relibility nd risk odels setting relibility requireents, rnfield University, UK. John Wiley & Sons, Ltd, 5. [3] A. ABOU JAOUDE, K. EL-TAWIL, S. KADRY, H. OURA, nd M. OULADSIE, Anlytic prognostic odel for dynic syste, Interntionl Review of Autotic ontrol (IREAO), oveber. [4] A. ABOU JAOUDE, S. KADRY, K. EL-TAWIL, H. OURA, nd M. OULADSIE, "Anlytic prognostic for petrocheicl pipelines", Journl of Mechnicl Engineering Reserch (JMER), April.
[5]. SUKUMAR, D.L. HOPP, nd B. MORA, "Extended finite eleent ethod nd fst rching ethod for three-diensionl ftigue crck propgtion", Engrg. Frcture Mech. 7 (3) 9-48. [6] J.. EWMA JR. nd I.S. RAJU, "An epiricl stress-intensity fctor eqution for the surfce crck", Engrg. Frcture Mech. 5 (/) (98) 85-9. [7]. OLQUHOU, "Ftigue Anlysis of n FEA Model of Suspension oponent, nd oprison with Experientl Dt", Sipson Interntionl UK Ltd, Hlifx, Englnd. [8] J. DRAPER, Sfe Technology Liited, Sheffield, Englnd. [9].E. FROST nd D.S. DUGDALE, "Ftigue Tests on otched Mild Steel Pltes with Mesureents of Ftigue rcks", Journl of the Mechnics nd Physics of Solids 5:8-9, 957. [].E. FROST, "otch Effects And The riticl Alternting Stress Required To Propgte A rck In An Aluinu Alloy Subject To Ftigue Loding", Journl of Mechnicl Engineering Science, 9-9, 96. [] F.A. OLE nd T.H. TOPPER, "Overstrin Effects During Vrible Aplitude Service History Testing", Interntionl Journl of Ftigue, Vol, o.3, pp3-36, 98. [] D.L. DUQUESAY, M. A. POMPETZKI, nd T. H. TOPPER, "Ftigue Life Prediction for Vrible Aplitude Strin Histories", SAE Pper 934, Society of Autootive Engineers. [3] J. LEE, "Srt Products nd Service Systes for E-business Trnsfortion", 3e onférence Frncophone de Modélistion et Siultion «onception, Anlyse et Gestion des Systèes Industriels» MOSIM, du 5 u 7 vril, Troyes (Frnce), 4. [4] K. EL-TAWIL, Mécnique Alétoire et Fibilité, ours de Mster R Mécnique, EDST, Université Libnise, 4. [5] K. EL-TAWIL, S. KADRY, Ftigue Stochstique des Systèes Mécniques Bsée sur l Technique de Trnsfortion Probbiliste, Internl report, Lebnese University, grnt reserch progr,. [6] F. VAKILI-TAHAMI, M. ZEHSAZ, nd M.R. ALIDADI, "Ftigue nlysis of the weldents of the suspension-syste-support for n off-rod vehicle under the dynic lods due to the rod profiles", Asin Journl of Applied Sciences ():-, Mlysi, 9. [7] T.K. SERGEEVA, A.. BOLOTOV, et l., "Monitoring of steel condition in in pipelines during their stress-corrosion induced filures", heicl nd oil Mchinery, o., pp. 7-76, 996. [8].E. JASKE, "Fitness-for-service ssessent for pipelines subject to stress-corrosion crcking", The Pipeline Pigging nd Integrity onference, Februry. 3
[9] A. ABOU JAOUDE, K. EL-TAWIL, S. KADRY, H. OURA, nd M. OULADSIE, "Prognostic odel for buried tubes", Interntionl onference on Advnced Reserch nd Applictions in Mechnicl Engineering (IARAME'), otre De University, Louize, Lebnon, June 3-5,. [3] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, nd M. OULADSIE, "Lifetie nlytic prognostic for petrocheicl pipes subject to ftigue", SAFEPROESS, 8th IFA Syposiu on Fult Detection, Supervision nd Sfety of Technicl Processes, Mexico ity, Mexico, August 9-3,. [3] J.B. LIGO nd G.R. MAYER, Buried pipes, Mechnicl Mechnics Dept., Michign, Technologicl University, Houghton. [3] WAIMAKARIRI, District council, Designing for Surge & Ftigue, Stndrd Specifiction, QP-84, Issue, 8. [33] H. S. DA OSTA MATTOS, E. M. SAMPAIO, nd R. M. ALVES ORTES, 7, "Anlysis of coposite sleeve reinforceent systes for etllic pipelines with loclized iperfections or dge", Mechnics of Solids in Brzil. [34]. BROW, Intrinsic Lifetie of Polyethylene Pipelines, University of Pennsylvni, Deprtentl pper, Polyer Engineering nd science, vol 74, 7. [35] J. ADERSO, Design nd Instlltion of Mrine Pipeline, Blckwell edition, 5. [36] Y. BAI nd Q. BAI, Subse Pipelines nd Rivers, Elsevier, 5. [37] A.. PALMER nd R. A. KIG, Subse Pipeline Engineering, publisher Pennwell Book 4
HAPTER III AALYTI OLIEAR PROGOSTI MODEL OF DYAMI SYSTEMS 5
III. - Introduction Until now, dges hve been ssued to ccuulte linerly (Miner s lw) even though it is unlikely to be the cse of brittle teril. The present chpter intends to develop ore dvnced prognostic tool by exploring the nonliner side of cuultive dge. This is in order to tke into ccount the nture nd the ode of pplied constrints nd influent environent tht cn ccentute the nonliner spect relted to soe terils behvior subject to ftigue effects. In hpter II we hve considered the clssicl cse of liner dge ccuultion clled Miner's lw [] widely used in specilized literture for ost steel terils. In the present chpter we will explore the nonliner cse of dge cuultive lw to tke into ccount the rel behvior of soe terils subject to ftigue ctions, especilly when the nture of pplied constrints nd influent environent contribute to plify the nonliner spect of dge. Its iportnce is cler since s we know it is not very well treted until now. In ddition to this, the intended stochstic study, subject of hpter IV, needs to consider this nonlinerity in cuultive dge. Figures 3. nd 3. represent n exple of liner nd nonliner dge ccuultion lws [,3]. Where n nd n re the nuber of loding cycles, R nd R re respectively the criticl nuber of cycles for the loding levels ε nd ε, nd t is the tie of loding. These two figures show the influence of loding order between liner nd nonliner cses; in fct, when sll loding ε precedes high loding ε (upper cse) the liner rule (Plgren-Miner) does not ke the difference for this order wheres the nonliner rule perits to give convex curve of dge (figure 3.) which cn be odeled by double liner dge rule (DLDR) (figure 3.) (refer to prgrph III.3). When high loding ε precedes sll loding ε (lower cse) lso the liner rule is insensible to this order contrrily to the nonliner rule where it gives concve curve of dge (figure 3.) odeled in soe ethods by double liner dge rule (DLDR) (figure 3.) (refer to prgrph III.3). 6
Plgren- Miner rule Figure 3. - Liner dge ccuultion. Figure 3. - onliner dge ccuultion. III. - Stte-of-the-Art: onliner Dge Accuultion The subject of cuultive ftigue dge is extreely coplex, nd vrious theories hve been proposed like in reference [4] to predict ftigue life in dvnce of service. The ost widely known nd used procedure is the liner dge rule coonly clled the Miner rule. The liner dge rule, which indictes tht sution of cycle rtios is equl to unity, is not copletely ccurte; however, becuse of its siplicity nd becuse of its greeent with experientl dt for certin cses it is frequently used in design. If new ethod is to replce the liner dge rule in prcticl design, uch of the siplicity of the liner dge rule ust be retined. For exple, the double liner dge rule (DLDR) explined lter, retins uch of this siplicity nd t the se tie ttepts to overcoe soe of the liittions inherent in the conventionl liner rule. One of the liittions of the liner dge rule is tht it does not consider the effect of order of loding. For exple, in two-stress-level ftigue test in which high lod is followed by low lod, the cycle rtio sution is less thn, wheres low lod followed by high lod produces cycle rtio sution greter thn. The effect of residul stress is lso not properly ccounted for by the conventionl liner dge rule, nor does it consider cycle rtios pplied below the initil ftigue liit of the teril [4]. Since prior loding cn reduce the ftigue liit, cycle rtios of stresses pplied below the initil ftigue liit should be ccounted for [4]. 7
In ddition, coxing effects present in soe strin-ging terils [4] in which the pproprite sequence of loding y progressively rise the ftigue liit re not ccounted for by the liner dge rule. Vrious ethods hve been proposed s lterntives to the liner dge rule. one overcoes ll the deficiencies, nd ny introduce dditionl coplexities tht either preclude or ke their use extreely difficult in prcticl design probles. Ftigue dge increses with pplied lod cycles in cuultive nner which y led to frcture. uultive ftigue dge nlysis plys key role in life prediction of coponents nd structures subjected to fields lod histories. Since the introduction of dge ccuultion concept by Plgren-Miner, the tretent of cuultive ftigue dge hs received incresingly ore ttention. As result, ny dge odels hve been developed. Even though erly theories on cuultive ftigue dge hve been reviewed by severl reserchers, no coprehensive report hs ppered recently to review the considerble efforts de since the lte 97s. A generl cuultive dge ethodology is derived fro the bsic reltion specifying crck growth rte (increent) s power lw function of the stress intensity fctor. The crck is llowed to grow up to the point t which it becoes unstble, thereby deterining the lifetie of the teril under the prescribed stress progr. Dge ccuultion in terils is very iportnt, but very chllenging to chrcterize in eningful nd relible nner. As the possible dge ccuultes, the reining lifetie under future lods becoes ore liited. The ultite gol is to be ble to predict the reining lifetie s the pst history of loding induces growing stte of dge. More succinctly, the coon purpose is to be given coplete loding spectru nd then predict how fr into the loding sequence the teril cn rein coherent before suffering ctstrophic filure. The ost coon pproch to such probles is to recognize tht crcks under ftigue conditions usully grow in nner with the rte of growth expressed s stress level (stress intensity fctor) to soe exponent. This is widely known s the Pris lw nd hs been verified for ny terils over ny decdes of chnge on log scles. This power lw for is then used to predict the nuber of lod cycles until the crck reches pre-selected, 8
uncceptble size. Prticulr odels relte the rte of crck growth to nonliner functions of the stress intensity fctor. Another generl pproch is tht of Liner uultive Dge, LD. In this ethod increents of dge, expressed s frctions of lifetie t prticulr stress levels, re linerly dded together to express totl dge nd thereby the lifetie (Plgren-Miner Lw). The ethod is copletely epiricl, but quite widely used becuse of its siplicity nd utility. However, LD is widely cknowledged to be indequte. This is prtilly bsed upon its epiricl nture nd prtly bsed upon its prediction of unstisfctory results [5]. Miner's rule ssues tht dge contribution fro ech cycle of the loding history is independent fro the other cycles. Therefore, the dge inflicted by n stress cycles with defined gnitude S is given by: D n () Where denotes the cycles to filure t S fro the constnt-plitude S- curve (WÖhler curve). For ll stress levels this dge rules yields []: D i d i i ni i () Where n i is the nuber of cycles hving plitude S i. In the LD, the esure of dge is siply the cycle rtio with bsic ssuption of constnt work bsorption per cycle, nd chrcteristic ount of work bsorbed t filure. The energy ccuultion, therefore, leds to liner sution of cycle rtio or dge. The in deficiencies with LD re its lod-level independnce, lod sequence independnce nd lck of lod-interction ccountbility. Howerver, due to the inherent deficiencies of the LD, no tter which version is used, life prediction bsed on this rule is offen unstisfctory. Experientl evidence under copletely reversed loding condition often i indictes tht d > for low-to-high (L-H) loding sequence, nd d < for i 9
high-to-low (H-L) loding sequence. To reedy the deficiencies ssocited with the LD, soe uthors like in reference [6] introduced the concept of dge curves nd speculted tht these curves ought to be different t different stress-levels. Then the first nonliner loddependent dge theory ws proposed by Mrco nd Strkey [6], it is represented by power reltionship D i d i α where α i is vrible quntity relted to the i th loding level. The plots of these curves re shown in figure 3.3. In this figure, digonl stright line represents the Miner rule which is specil cse of the bove eqution () with α. As illustrted by figure 3, life clcultions bsed on Mrco-Strkey theory would result in di >for L-H lod sequence, nd in i Dge - D > > 3 For opertion t followed by opertion t 3 ni ( AB + D) < i, i For opertion t 3 followed by opertion t ni ( AB + ED) > i, i ycle Rtio n i / i Miner's rule.5 A E B D Figure 3.3 - Schetic representtion of dge versus cycle rtio for the Mrco-Strkey theory. 3 i d < for H-L lod sequence. III.. - Dge Theories Bsed on Endurnce Liit Reduction On the other hnd, the concept of chnge in endurnce liit due to pre-stress exerted n iportnt influence on subsequent cuultive ftigue dge reserch. Koers nd Bennett [6] further investigted the effect of ftigue prestressing on endurnce properties using two-level step loding ethod. Their experientl results suggested tht the reduction in endurnce strength could be used s dge esure, but they did not correlte this dge preter to the life frction. This type of dge odels bsed on endurnce liit reduction re non-liner nd ble to ccount for the lod sequence effect. Soe of these odels cn lso be used for predicting the instntneous endurnce liit of teril, if the loding history is known. one of these odels, however, tke into ccount lod interction effects.
III.3 - onliner-dge-bsed Prognostic Vrious pproches to prognostics hve been developed tht rnge in fidelity fro siple historicl filure rte odels to high-fidelity physics-bsed odels like in reference [7]. The required infortion (depending on the type of prognostics pproch) include: engineering odel nd dt, filure history, pst operting conditions, current conditions, identified fult ptterns, trnsitionl filure trjectories, intennce history, environent of equipent, syste degrdtion nd filure odes. A nuber of different ethods hve been pplied to study prognosis of degrded coponents. In generl, prognostics pproches cn be clssified into three priry ctegories: () Model driven, () Dt driven, (3) And probbility-bsed prognostic techniques. The in dvntge of odel bsed pproches is their bility to incorporte physicl understnding of the onitored syste. In ddition, in ny situtions, the chnges in feture vector re closely relted to odel preters nd functionl pping between the drifting preters nd the selected prognostic fetures cn be estblished []. Moreover, if the understnding of the syste degrdtion iproves, the odel cn be dpted to increse its ccurcy nd to ddress subtle perfornce probles. onsequently, they cn significntly outperfor dt-driven pproches. But, this closed reltion with theticl odel y lso be strong wekness: it cn be difficult, even ipossible to ctch the syste's behvior. Further, soe uthors think tht the onitoring nd the prognostic tools ust evolve s the syste does. An erlier proposed procedure [8] (hpter II) belongs to the first prognostic pproch. It is bsed on physicl odel nd leding to norlized degrdtion indictor. It is focused on developing nd ipleenting effective dignostic nd prognostic technologies with the bility to detect fults in the erly stges of degrdtion. Erly detection nd ccurte nlysis y led to better prediction nd end of life estites by trcking nd odeling the degrdtion process.
The ide ws to use these estites to ke ccurte nd precise prediction of the tie to filure of coponents. The chosen filure ode ws the ftigue filure forulted theticlly on the bse of nlytic dge lws of Pris nd Miner. The lst lw is liner cuultive dge odel (figure 3.). Even tht these lws re very well known in echnics of rupture but their uses in the present prognostic procedure help s support for n exple of degrdtion expression. Pst reserch hs shown there is nonliner interction effect between high cycle ftigue (HF) nd low cycle ftigue (LF) in ny engineering terils. This effect hs been observed within unixil lodings, but is often ore pronounced under ultixil loding, prticulrly when the loding is non-proportionl. An exple here is the developent of ftigue dge ssessent ethods for turbine engine terils cobining the LF nd HF cycles. The nonliner interction effect precludes the use of the ost coon technique for liner dge ccuultion. A thorough review of nonliner cuultive dge (figure 3.) ethodologies [9] shows tht these techniques hve included siple extensions of the liner dge rule to include nonliner ters. Severl nonliner ethods exist, including endurnce-liit odifiction techniques, frcture-echnics bsed pproches, continuudge, nd life-curve pproches. Trditionl ethods of dge sution hve been shown to provide n inccurte life prediction when ultiple lod levels re siultneously considered. This is due to the effect tht one lod level hs on the other(s). In the present study, the effect of HF loding hs hd ore detrientl effect when coupled with the LF lodings thn predicted by liner sution rule. onliner dge ccuultion theories cn ccount for this influence nd hve shown n iproveent in prediction. The stress levels were chosen to correspond to levels previously tested to filure, resulting in ftigue lives rnging fro pproxitely 5 to 7 cycles. A nonliner dge sution is required to properly define the ftigue process since the liner sution of dge is often not dequte to predict the service life of coponent when subjected to vrible-plitude lodings.
III.3. - Disdvntges of Liner Dge Accuultion The ost coon ethod of suing dge for loding spectru is the Plgren- Miner liner dge rule [] (figure 3.4). It is redily understood nd esy to ipleent nd is, therefore, the foundtion for ny of the other cuultive dge theories tht hve been proposed. Idelly, the sution of life rtios uultive dge D Relible Filure n i / i would equl one t filure. Figure 3.4 - Plgren-Miner's liner rule of dge. However, pst experients hve yielded rnge of rtios fro.7 to. for unixil lodings, resulting in filure predictions erring just slightly on the side of non-conservtive to ore thn the double for conservtive prediction []. For the bixil lodings, Miner's sution of.9 ws found in these experients [], indicting thus extreely nonconservtive results s it is so fr fro filure point (equl to.). This proves the dependence of Miner' lw on the lod directions. Also, the lrgest drwbck of the liner dge rule is its inbility to ccount for the order of loding. Tht is, the resulting filure prediction is independent of the lod interction effects tht hve been observed between high-cycle nd low-cycle lodings. It is this shortcoing tht hs propted the developent of severl nonliner cuultive dge theories. Hence, different non-liner dge rules hve been proposed in literture nd presented s follows. III.3. - Double Liner Dge Rule (DLDR) The current for of the DLDR ws proposed in 966 []. Insted of single stright line, set of two stright lines tht converged t coon "Kneepoint" would be used (figure 3.5). It helps differentite between the dge Kneepoints Loding phse Loding phse cused by the LF nd HF for ulti-level lodings. Its bsis is the replceent of the continuous dge curve by two stright lines. Ech liner phse cn be nlyzed by Figure 3.5 - Double Liner Dge 3
Plgren-Miner liner dge rule. The difficulty encountered when utilizing the DLDR is estblishing the loction of the trnsitory point between the two loding phses (eqution 3). The DLDR is represented by the equtions (3) illustrted in figure 3.5. These equtions perit to clculte the dge ccuultion t ech loding cycle with respect to the double liner dge rule. n. 35 knee α n. 65 knee α (3) Where, n / nd n / re loding phses, α: teril preter. III.3.3 - Dge urve Approch (DA) To better describe ftigue filure using nonliner dge, insted of stright line, single continuous curve reflects ore ccurtely the influence of the loding (figure 3.6). For HF loding significnt nuber of cycles hd to be pplied before enough dge could ccuulte to cuse reduction in life. Once the pproprite nuber of cycles hd been pplied, the dge continued to ccuulte t n ever-incresing rte nd filure ws soon to follow. For LF lodings, this behvior ws less pronounced. Figure 3.6 - Dge urve Approch. A workble eqution bsed on erly crck growth theories ws provided [3]: D n f α f ref (4) The ipleenttion of the DA odel is illustrted in figure 3.5. The priry dvntge in eploying the DA odel lies in its bility to crete identicl dge curves 4
for different life references. The liner dge line becoes the reference life tht is used to estblish the teril constnt in (4), nd other dge curves shift vlues ccordingly. Where, D is the dge ccuulted, n /, n /, nd n 3 / 3 re the loding phses, f is the criticl nuber of cycles, ref is the reference nuber of cycles (reference life). III.3.4 - Double Dge urve Approch (DDA) Although the DA shows lrge potentil in ccurtely predicting filure in ulti-level loding, there is one serious drwbck when considering high-low loding. It cn be seen upon exintion tht with the ppliction of just few high-plitude cycles, there is rpid decrese in reining life t the low-plitude lod level. This result is fro lck of the low-rnge dt needed to djust the shpe of the curve during the odels conception. To iprove the odel, Mnson nd Hlford [4] included liner ter to shift the curves wy fro the x-xis. The difficulty would be to llow this new ter to hve significnt influence t low life rtios but negligible effect t higher rtios. The resulting double dge curve pproch (DDA) closely pproxited the DLDR in the lower-life regie nd the DA in the higherlife regie, where ech odel perfored best. The eqution for the DDA is shown in (5): D n q γ + γ [ q ] n γ ( q ) γ (5) Where, D dge ccuulted, n nuber of pplied cycles t given lod level, nuber of cycles required to fil t the se lod level s n. Figure 3.7 - Double Dge urve Approch. 5
.35 q nd q α ref.65 ref re preters, γ 5 is constnt representing two intersecting stright lines which cn be replced by single curve, α α, β re teril dependent preters tht ust be experientlly deterined (typiclly tken s.5 nd.4, respectively). ref β The DDA odel is illustrted scheticlly in figure 3.7. otice the liner dge ccuultion t lower life rtios nd curviliner dge ccuultion t higher life rtios. otice tht the DDA odel is generl for which cn be pplied to wide rnge of terils nd equipents. III.4 - onliner uultive Dge Model The dge odel proposed in this chpter, whose evolution is up to the point of cro-crck initition, is represented in figure 3.8. The stte of dge of specien t prticulr cycle during ftigue is represented by sclr dge function D(). The gnitude D corresponds to no dge, nd D corresponds to the ppernce of the first crocrck (totl dge). D() D Filure Relible Figure 3.8 - onliner lw of dge. The following odel is chosen for the nonliner prognostic study. It represents the nonliner evolution of dge D in ters of the nuber of cycle given under the following first order nonliner ordinry differentil eqution [5]: dd d / ( D) α if / > if / < (6) 6
7 Where, : the nuber of cycles t filure s norlizing constnt, : the stress rnge in loding cycle, : the endurnce liit, it is function of the stress en in cycle: / where ; () ) ( < ult ult : the ultite tensile strength of the teril, nd α : they re constnts depending on the teril nd the loding condition (.9 nd α.3). This nonliner ordinry differentil eqution (6) needs to be solved in order to find n expression of D(). III.4. - Solution of the Differentil Eqution of Degrdtion The solution of the differentil eqution (6) is presented s follows: ( ) < > / if / if / α D d dd ( ) d dd D D D / α ( ) ( ) ( ) [ ] D D D D D + + + + + / ) ( / α α α α α ( ) ( ) ( ) ( ) ( ) ( ) / ) ( / ) ( + + + + + + α α α α α α D D D D ( ) ( ) ( ) ( ) ( ) ) ( / ) ( / ) ( + + + + + + α α α α α α D D D D (7)
8 Where, ) ( D D is the dge t cycles corresponding to n initil crck length. We choose n equivlent dge preter, to be esured by structurl helth onitoring. The plotting of expression (7) of D() is presented in figure 3.9. Figure 3.9 - onliner D() curve. Prticulr cse:, for Tke D ( ). / where ; ) ( α α + + D Filure cse: At filure we hve nd ) ( D, then eqution (8) gives: ( ) + + α α ( ) ( ) + + α α (8)
9 ( ) : Assue tht + + + α α α Therefore: ( ) + α III.4. - Reltion between D nd t Specific ycle Let us study the reltion between the degrdtion D nd the cycle of stress. To do tht esily let us integrte the reltion of degrdtion between cycle nd cycle ssuing tht filure occurs t cycle. Fro eqution (6), it cn be deduced tht: ( ) d dd D / α ( ) d dd D D D / α It cn be inferred lso: ( ) D D D D + + + + + + / ) ( / ) ( / α α α α α α ( ) ( ) / / + + + + α α α α D D ( ) / + + α α D (9)
Then: III.4.3 - Recursive Reltion of onliner Dge D To construct recursive reltion for the sequence of D, the procedure is s follows: ( ) ( ) ( ) ( ) ( ) [ ] D D D D D D D d dd D d dd D + + + + + + + + + + α α α α α α α / where ; / The previous recursive reltion leds to sequence of vlues D whose liit is D :,,,,,,, + D D D D D D And s the stress-lod is expressed in ters of tie (t), then we cn plot the curve of degrdtion D in ters of tie (t). Therefore, our prognostic odel in the nonliner cse is given by: ( ) ) ( + + + + α α α D D ( ) / ) ( + + α α D ( ) ( ) ( ) ) ( ) ( + + + + + + + + α α α α α α D D D D () () ()
III.5 - Appliction to Suspension Syste Reconsider the exple of hpter II concerning the vehicle suspension syste nd pply the nonliner odel of dge developed in prgrph III.4.3 (eqution ) in order to clculte the prognostic of this syste. The following preters re considered in the siultion [6]: is norlizing constnt tken to be equl to the nuber of cycles t filure ( 7 ) α estited to be.3,.9, / is the stress lod plitude in one cycle, this preter is generted s n input lod resulting fro the rod profile nd whose en is tken to be equl to 8 MP, is the ftigue liit (endurnce liit of the teril) tken to be equl to 8 MP. Tble 3. - Sttisticl chrcteristics of ech ode of rods. Rod Mode Severe (ode ) Fir (ode ) Good (ode 3) Men of x oefficient of Stndrd Lw j Vrition of Devition ( x j in ) x j in % (in ) 5% 5 orl 5 % 5 orl 5 5%.5 orl For ore detils bout the dt of this ppliction, refer to hpter II. III.5. - Results of the Siultion The siultions of the degrdtion of vehicle suspension subject to the severe, fir, nd good odes of rod profiles re represented respectively in figures 3., 3., nd 3..
Figure 3. - Suspension degrdtion under nonliner lw for severe ode of rod excittion. Figure 3. - Suspension degrdtion under nonliner lw for fir ode of rod excittion.
Figure 3. - Suspension degrdtion under nonliner lw for good ode of rod excittion. Figures 3.3 nd 3.4 represent respectively the evolution of degrdtion D nd of the RULs for the suspension for three odes of rods with profile properties indicted in tble 3.. Figure 3.3 - Suspension degrdtion under nonliner lw for three odes of rod excittions. 3
Figure 3.4 - Suspension RULs under nonliner lw for three odes of rod excittion. The RULs evlutions in figure 3.4 re deduced fro the expression -. In fct is the necessry cycle nuber to rech filure (ppernce of the first cro-crcks) nd is the cycle nuber corresponding to crck length. ote tht is the initil cycle nuber t the beginning tken generlly equl to. These curves decrese fro entire lifetie of the device to zero where D. Fro these curves we cn deduce t ech cycle RUL() of the device nd hence the prognostic result cn be inferred. The expected lifeties re s follows: Mode : 9,47,7 cycles Mode :,63,8 cycles Mode 3: 3 8,95,4 cycles III.5. - onversion of RUL into Yers To convert the suspension lifetie into yers' unit, knowing tht ech cycle's durtion is seconds, then: RUL(s) RUL(). If we ssue tht the suspension tie usge is % of dy (.4 hours/dy), then the expected lifeties' durtions re (refer to hpter II, Prgrph 3..): 4
For ode : 9,47,7(cycles) (s) 5.738 yers 6(s) 6(in).4(hours) 365(dys) For ode :,63,8(cycles) (s) 7.65 yers 6(s) 6(in).4(hours) 365(dys) For ode 3: 8,95,4(cycles) (s).476 yers 6(s) 6(in).4(hours) 365(dys) III.5.3 - oprison with the Liner se We cn deduce fro the two figures 3.5 nd 3.6 tht, first of ll, the nonliner cse of dge is ore optiistic nd ccurte thn the liner cse concerning the lifetie becuse the vlues re lrger. Secondly, the decresing of RULs in the nonliner cse is less steep t the end thn the liner cse becuse the nonliner curves rech the zero vlue progressively. 8 x 6 6 Reining Useful Lifetie RUL (cycles) 4 8 6 4 Mode 3 : Good Mode : Fir Mode : Severe..4.6.8..4 Degrdtion D Figure 3.5 - Liner cse. 6,836, cycles;,85, cycles; 3 7,, cycles. Figure 3.6 - onliner cse. 9,47,7 cycles;,63,8 cycles; 3 8,95,4 cycles. Finlly, we cn rerk tht ner the filure zone where D D the nonliner study sees to give here ore logicl nd relistic dge behvior for the different rod profiles thn the liner cse where the dge curves becoe identicl. In fct, between good 5
nd severe profiles, the nonliner cse kes the difference when pproching filure liit wheres the liner cse does not. The optiistic results obtined fro the nonliner cse cn be explined by the fct tht when the rel nonliner trends of degrdtion re of concve for then the dge ccuultion is overestited when using liner for (figure 3.7). Filure level Degrdtion index onvex Liner oncve Lifetie Figure 3.7 - Different degrdtion trends. Referring to the references [7,8,9,8], the vlidtion of the present results cnnot be explined without tking into considertion the nonliner bsis of the current study contrry to the liner dge odel dopted in the previous references. Therefore, the results got here re relistic when copred to those obtined by the works of these uthors. III.5.4 - Advntges of the Proposed Model In coprison with predictive RUL odels vilble in literture [], the dvntges of the present odel re: ) It is siple nd prcticl in ppliction to vrious industril systes for ftigue life prediction. b) The fct of using nonliner lw, if it exists, for dge ccuultion, kes it ore efficient nd relistic in predicting the reining useful lifetie. 6
c) When ultiple lod levels re siultneously considered, the liner lw of dges ccuultion like Miner's lw leds to inccurcy [] in life prediction wheres the nonliner lw of dge perits to consider the effect entioned bove. d) It tkes into ccount the lod interction effects between high-cycle nd low-cycle lodings contrry to predictive odels bsed on liner dge lw. e) Its efficiency reltively to other odels hs been often ore pronounced under ultixil loding, prticulrly when the loding is non-proportionl. f) It considers the influent environent tht cn ccentute the nonliner spect relted to soe terils behvior subject to ftigue effects (brittle terils for exple). g) The Pris' lw of ftigue for crck growth dopted in the present odel is siple to use nd requires two preters esily obtined. It is the siplest to perfor becuse no lod history hs to be considered. In fct, it llows n excellent prediction odel results for crck lives below 5 cycles. III.6 - Appliction to Pipeline Syste Reconsider the exple of hpter II concerning the pipeline syste nd pply the nonliner odel developed in prgrph III.4.3 (eqution ). The deterinistic tringulr siultion of the three odes of internl pressure is de using the preters given in tble 3.. Tble 3. - Sttisticl chrcteristics of ech pressure ode. Pressure Mode High (ode ) Middle (ode ) Low (ode 3) P j (MP) δ Pj (%) Lw 8 % Tringulr 5 % Tringulr 3 % Tringulr The study covers three types of pipes: unburied, buried nd offshore. 7
III.6. - Unburied Pipe se The cse studied here is tht of unburied pipes (in free ir). The siultions of the pipe degrdtion for high, iddle nd low odes of internl pressure re represented respectively in figures 3.8, 3.9, nd 3.. Figure 3.8 - Pipelines degrdtion under high ode pressure for nonliner lw cse (unburied pipes). Figure 3.9 - Pipelines degrdtion under iddle ode pressure for nonliner lw cse (unburied pipes). 8
Figure 3. - Pipelines degrdtion under low ode pressure for nonliner lw cse (unburied pipes). The degrdtion evolution (figure 3.) nd the RULs evolution (figure 3.) re obtined for ech ode of internl pressure in ters of exploittion tie nd degrdtion stte D. Figure 3. - Pipelines degrdtion under three odes of pressure for nonliner lw cse (unburied pipes). 9
Figure 3. - RULs evolution for pipelines under three odes of pressure for nonliner lw cse (unburied pipes). The expected lifeties deduced re s follows: Mode : 3.53 yers Mode : 6. yers Mode 3: 3.59 yers. III.6.. - oprison with the Liner se Figure 3.3 - Liner cse. 3.3 yers; 4.68 yers; 3 6.85 yers. Figure 3.4 - onliner cse. 3.53 yers; 6. yers; 3.59 yers. 3
It cn be deduced fro these two figures 3.3 nd 3.4 tht first of ll the nonliner cse of dge is slightly less conservtive thn the liner cse concerning the lifetie. Secondly, the decresing of RULs in the nonliner cse is less cute t the end thn the liner cse becuse the nonliner curves rech progressively the zero vlue. Finlly, we cn rerk tht ner the filure zone where D D the nonliner study sees to give here ore logicl nd relistic dge behvior for the different pressure vlues thn the liner cse where the curves coincide. In fct, we note in the nonliner cse cler difference between low nd high pressures when pproching filure liit wheres the liner cse does not. III.6. - Buried Pipe se In the cse of buried pipes (underground), the siultions of degrdtion under high, iddle, nd low odes of internl pressure re represented respectively in figures 3.5, 3.6, nd 3.7. Figure 3.5 - Pipelines degrdtion under high ode of pressure for nonliner lw (buried pipes). 3
Figure 3.6 - Pipelines degrdtion under iddle ode of pressure for nonliner lw (buried pipes). Figure 3.7 - Pipelines degrdtion under low ode of pressure for nonliner lw (buried pipes). We therefore obtin the degrdtion evolution (figure 3.8) nd the RULs evolution (figure 3.9) for ech ode of internl pressure in ters of exploittion tie nd degrdtion stte D. 3
Figure 3.8 - Pipelines degrdtion under three odes of pressure for nonliner lw (buried pipes) Figure 3.9 - RUL evolution for pipelines under three odes of pressure for nonliner lw (buried pipes). The expected lifeties deduced re s follows: Mode : 8.84 yers Mode : 5.3 yers Mode 3: 3 6.54 yers. 33
III.6.. - oprison with the Liner se Figure 3.3 - Liner cse. 8.33 yers;.87 yers; 3 7.35 yers. Figure 3.3 - onliner cse. 8.84 yers; 5.3 yers; 3 6.54 yers. We cn deduce fro the two figures 3.3 nd 3.3 tht first of ll the nonliner cse of dge is obviously less conservtive thn the liner cse concerning the lifeties. Secondly, the decresing of RULs in the nonliner cse is less cute t the end thn the liner cse becuse the nonliner curves rech progressively the zero vlue. Finlly, we cn notice tht ner the filure zone where D D the nonliner study sees to give here ore logicl nd relistic dge behvior for the different pressure vlues thn the liner cse where the curves coincide. In fct, we note in the nonliner cse cler difference between the different pressures when pproching filure liit wheres the liner cse does not. III.6.3 - Offshore Pipe se For offshore pipes (under se wter), the siultions of degrdtion under high, iddle, nd low odes of internl pressure re represented respectively in figures 3.3, 3.33, nd 3.34. 34
Figure 3.3 - Pipelines degrdtion under high ode of pressure for nonliner lw (offshore pipes). Figure 3.33 - Pipelines degrdtion under iddle ode of pressure for nonliner lw (offshore pipes). 35
Figure 3.34 - Pipelines degrdtion under low ode of pressure for nonliner lw (offshore pipes). We therefore obtin the degrdtion evolution (figure 3.35) nd the RULs evolution (figure 3.36) for ech ode of internl pressure in ters of exploittion tie nd degrdtion stte D. 36 Figure 3.35 - Pipelines degrdtion under three odes of pressure for nonliner lw (offshore pipes).
Figure 3.36 - Pipelines RUL evolution under three odes of pressure for nonliner lw (offshore pipes). The expected lifeties deduced re s follows: Mode :.9 yers Mode : 9. yers Mode 3: 3 33.67 yers. III.6.3. - oprison with the Liner se Figure 3.37 - Liner cse.7 yers; 4.84 yers; 3.69 yers. Figure 3.38 - onliner cse.9 yers; 9. yers; 3 33.67 yers. 37
We cn deduce fro these two figures 3.37 nd 3.38 tht, first of ll, the nonliner cse of dge is undoubtedly less conservtive thn the liner cse concerning the lifetie. Secondly, the decresing of RULs in the nonliner cse is less steep t the end thn the liner cse becuse the nonliner curves rech progressively the zero vlue. Finlly, we cn notice tht ner the filure zone where D D the nonliner study sees to give here ore logicl nd relistic dge behvior for the different pressure vlues thn the liner cse where the curves coincide. In fct, we cn see in the nonliner cse cler difference between the different pressures when pproching filure liit wheres the liner cse does not. III.6.4 - Vlidtion of the Pipelines Lifeties Referring to the references [,,3], the present results of pipelines nonliner dge odel re relistic when copred to those obtined by the works of these uthors. In coprison with the liner odel, the lifeties in the nonliner cse re ore ccurte nd ore econoic since they led to lrger intennce intervls. In fct, the typicl lifetie of offshore pipes is 5 yers on verge [3] which is very close to the lifeties' verge for the offshore pipes obtined by the nonliner siultion odel:.9 yers (for ode) + 9. yers (for ode ) + 33.67 yers (for ode 3).3 yers 3. Moreover, the design procedures for offshore pipelines re still under developent which hs led to substntil field of reserch tht dels with proper physicl deterintion of the ny spects of pipeline life cycle. In generl, ny different spects before nd during the life cycle of pipeline ust be considered. In fct, plnning dends gret del of considertions. During the life cycle fro fbriction to bndoning the instlled pipeline fter yers of opertion, the pipeline ust provide sfe trnsporttion. Therefore, in cse of filure, severe environentl pollution nd gret econoic loss y occur. 38
III.7 - onclusion In this chpter, the nonliner spect of dge ccuultion is introduced in the developed odel t the plce of the liner ccuultion of Miner. It llows tking into ccount the ultixil loding, prticulrly when the loding is non-proportionl, nd nonliner interction effect exists between LF nd HF loding cycles. Fro the resolution of first order nonliner ordinry differentil eqution relting the degrdtion to the nuber of cycles, we deduce recursive reltion between two consecutive degrdtion esures beside the environentl nd teril preters. The deduced reltion constitutes the nonliner prognostic odel. This dvnced prognostic odel is pplied to study the lifetie of two systes in siultion: the suspension coponents nd the petrocheicl pipelines in their three odes. The results of prognostic studies show tht the nonliner study gives ore logicl nd relistic dge behvior for the different loding vlues thn the liner cse. In fct, we note in the nonliner cse cler difference between two extree lodings when pproching filure liit wheres the liner cse does not. The nonliner cse study of suspensions shows optiistic results explined by the fct tht when the rel trends of degrdtion hve concve shpe then the dge ccuultion is overestited when using liner shpe. 39
References [] M.A. MIER, uultive Dge in Ftigue, Journl of Applied Mechnics, vol., A59-A64, 945. [] J. LEMAITRE nd R. DESMORAT, Engineering Dge Mechnics, Springer, 5. [3] Z. HASHI nd A. ROTEM, A uultive Dge Theory of Ftigue Filure", Mts. Sci nd Eng., 34, pp. 47-6, 978. [4]. BYIGTO, M. ROEMER, G. KAPRZYSKI, nd T. GALIE, "Prognostic Enhnceents to Dignostic Systes for Iproved ondition-bsed Mintennce", in: Proc. of IEEE Aerospce onference,. [5] S.S. MASO, J.. FREHE, nd.r. ESIG, "Appliction of double liner dge rule to cuultive ftigue, Lewis reserch center", Presented t Syposiu on rck Propgtion sponsored by the Aericn Society for Testing Mterils, Atlntic ity, ew Jersey, June 6 to July, 966. [6] R.M. HRISTESE, "A Physiclly Bsed uultive Dge Forlis", Lwrence Liverore tionl Lbortory nd Stnford University, 7. [7] A. FATEMI nd L. TAGT, Deprtent of Mechnicl, industril nd Mnufcturing Engineering, The University of Toledo, Toledo, OH 4366, USA, June 997. [8] J. LUO, M. AMBURU, K. PATTIPATI, L. QIAO, M. KAWAMOTO, nd S. HIGUSA, "Model-Bsed Prognostic Techniques", in: Proc. of IEEE Autotestcon, pp. 33-34, 3. [9] A. ABOU JAOUDE, K. EL-TAWIL, S. KADRY, H. OURA, nd M. OULADSIE, Anlytic Prognostic Model for Dynic Syste, Interntionl Review of Autotic ontrol (IREAO), oveber. [] E. GOODI, A. KALLMEYER, nd P. KURATH, "Evlution of onliner uultive Dge Models for Assessing HF/LF Interctions in Multixil Lodings", Reserch report, University of Dyton Reserch Institute, 7. [] J. SHIGLEY nd. MISHKE, Mechnicl Engineering Design, 5 th ed. p. 3. McGrw-Hill, Inc., 989. [] S.S. MASO, J.. FREHE, nd.r. ESIG, "Appliction of Double Liner Dge Rule to uultive Ftigue", ASTM STP 45, pp. 384-4, 967. [3] S.S. MASO nd G.R. HALFORD, "Prcticl Ipleenttion of the Double Liner Dge Rule nd Dge urve Approch for Treting uultive Ftigue Dge", Interntionl Journl of Ftigue. vol. 7, pp. 69-9, 98. [4] G.R. HALFORD nd S.S. MASO, "Reexintion of uultive Ftigue Dge Lws", Structure Integrity nd Durbility of Reusble Spce Propulsion Systes, ASA P- 38, pp. 39-45, 985. 4
[5] S.S. KULKARI, L. SU, B. MORA, S. KRISHASWAMY, nd J.D. AHEBAH, "A Probbilistic Method to Predict Ftigue rck Initition", Interntionl Journl of Frcture (6) 37:9-7, Springer, 6. [6] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, nd M. OULADSIE, "Anlytic nd onliner Prognostic for Vehicle Suspension Systes", IEEE Interntionl onference on Prognostic nd Helth Mngeent (PHM ), Denver, olordo, USA, June 8-,. [7]. SAKAVARAM et l., "Model-bsed nd dt-driven prognosis of utootive nd electronic systes", 5 th Annul IEEE onference on Autotion Science nd Engineering, Bnglore, Indi, August -5, 9. [8] J. WRE, "Ftigue nd durbility testing", Prosig oise nd Vibrtion Mesureent Blog, My 6. [9] Z. HUSI, M.M. RAHMA, K. KADIRGAMA, M.M. OOR, nd R.A. BAKAR, "Prediction of ftigue life on lower suspension r subjected to vrible plitude loding", tionl onference in Mechnicl Engineering Reserch nd Postgrdute Studies, nd MER, pp. -6, Mlysi, Deceber. [] S.M. BEDE, S. ABDULLAH, nd A.K. ARIFFI, "Review of ftigue crck propgtion odels for etllic coponents", Europen Journl of Scientific Reserch, vol. 8, o.3, pp. 364-397, EuroJournls Publishing, Inc., 9. []. HUAG, Structurl Helth Monitoring Syste for Deepwter Risers with Vortexinduced Vibrtion: onliner Modeling, Blind Identifiction Ftigue/Dge Estition nd Locl Monitoring using Mgnetic Flux Lekge, A Thesis Subitted in Prtil Fulfillent of the Requireents for the Degree Doctor of Philosophy, Mechnicl Engineering nd Mteril Science, Houston, Texs, June. [] DV, Recoended Prctice Det orske Verits Dnv-Rp-F7, Risk Assessent of Pipeline Protection, October. [3] K. RUBY nd P.A. HARTVIG, Free-spn Anlyses of n Offshore Pipeline, Thee: Design nd Anlysis of Advnced/Unusul Structures, Deprtent of ivil Engineering, Alborg University, 8. 4
HAPTER IV STOHASTI LIEAR AD OLIEAR AALYTI PROGOSTI MODEL 43
IV. - Introduction In our nlyticl odel of hpter II, dges hve been ssued to ccuulte linerly (using Miner s lw) even though it is unlikely to be the cse of brittle teril. Afterwrd nonliner cuultive dge is explored in hpter III [] to tke into ccount the level nd the ode of the pplied constrints nd influent environent tht cn ccentute the nonliner spect relted to soe terils behvior subject to ftigue effects. Other resons cn disturb the prediction cpcity of the odel which is the fluctutions of soe bsic preters; these fctors cn be tken into ccount by dopting stochstic odeling. In the present chpter, stochstic nlysis is introduced in ddition to the previous nonliner odel in order to ke it ore ccurte in the RUL prediction. It is done by considering soe preters s rndo vribles []. Our i is to ke the odel generl prognostic tool tht cn be cpble of well predicting the RUL of syste bsed on n nlyticl liner nd nonliner dge ccuultion in either deterinistic or stochstic context. Knowing tht the RUL cn be expressed in ftigue by ens of vrious fors like: criticl crck length or criticl nuber of loding cycles or teril tencity K I fro which we cn write vrious liit sttes or perfornce criteri. IV. - Stte-of-the-Art: Stochstic Ftigue Modeling There is significnt interest in iproving our understnding of ftigue relted dge nd prediction of the useful residul life of coponents experiencing ftigue dge. One of the principl tools for odeling ftigue dge is liner elstic frcture echnics, nd the resulting odels hve fcilitted the design of ftigue resistnt echnicl nd erospce structurl coponents [3]. Decision tools for filure prognostics ust hve the cpbility to incorporte teril dge under both norl nd pek operting conditions [3,4]. The science nd technology of prognosis nd structurl helth ngeent offer the potentil for significnt enhnceents in the sfety, relibility nd vilbility of high-vlue resources [5,6]. This concept is bsed on closed-loop process whose successful 44
ipleenttion depends on the integrtion of severl ulti-disciplinry eleents including [7]: ) Onbord sensing of opertionl preters nd teril dge sttes; ) Dignosing trends, fult conditions, nd underlying dge; 3) Predicting reining useful life in ters of probbility of filure nd liits on relible perfornce; 4) And deciding upon pproprite courses of ction: whenever or not the resource is cpble of perforing given ission, or lterntively, is in need of inspection, intennce, or replceent. onsiderble uncertinty exists in the usge nd sensor inputs, s well s the required odeling nd ssocited teril property inputs. onsequently, there is n inherent need for the resoning eleent of the prognosis syste to be probbilisticlly-bsed. opleenting the vriety of onbord sensors re trditionl helth onitoring softwre tools for pttern recognition, neurl networks, Byesin updting, expert systes, nd fuzzy logic. The dvntge of these tools is tht, when properly pplied, they re highly efficient nd thus enble to onbord onitoring nd rel-tie dt fusion nd interpoltion. However, the disdvntge of these tools is tht they rrely involve considertion of the underlying physicl processes. onsequently, they require considerble epiricl clibrtion or "trining" for ech specific ppliction of interest. In contrst, probbilistic life prediction is typiclly bsed on teril property dt, finite eleent therl nd stress nlysis, pre-service inspection nd in-service onitoring for defects, nd dge ccuultion lgoriths. The dvntge of this pproch is tht it is ore enble to linkge with the underlying physicl echniss of dge (i.e., crck nucletion nd growth). Thus, the process is inherently suitble for extension into terils prognosis, novel concept tht seeks to cobine infortion on the teril dge stte with echnisticlly-bsed predictive odels. The fundentl gol of ll of these pproches is to fcilitte better-infored decisions whether for ission plnning in the field (over the short ter), or sustinent t the depot (over the longer ter). In fct, the optiu prognosis syste is likely to be soe 45
cobintion of trditionl dt-driven ethods nd probbilistic echnics ethods. Thus, in ny respects the bove tools cn be viewed s being copleentry. Probbilistic nlyses of prognostic uncertinty were perfored using probbilistic life prediction code DARWI [8,9] s deonstrtion pltfor. DARWI integrtes finiteeleent stress nlysis results, frcture-echnics-bsed life ssessent for low-cycle ftigue, teril noly dt, probbility of noly detection, nd inspection/onitoring schedules to deterine the probbility-of-frcture of rotor disks s function of operting cycles. In the study on lives of turbine engines [7], enhnceents were dded to the DARWI code to enble the type of nlyses required for prognosis: ) Estblishent of interfce with engine sensor dt; ) Adding of the ftigue crck initition nlysis to existing ftigue crck propgtion nlysis; 3) Incorportes the integrtion of crck initition nd propgtion lgoriths; including correltion effects between the two dge processes; 4) Adding dge-bsed lod filtering ethod to reduce coputtionl tie; 5) pbility to nlyze lrge nuber of inspections (or interrogtion up to once per flight cycle) to siulte continuous onitoring with n on-bord sensor. Although DARWI contins severl probbilistic solutions ethods, the nlyses in reference to [7] were perfored using Monte rlo siultion. Other odels hve been proposed to describe the rndo behvior of ftigue crck growth in etls. In Yng nd Mnning s stochstic odel [,], siple second order pproxition of deterinistic crck growth odel is used with rndo coponent. An experientl study ws conducted by Wu nd i [,3] using this concept, which confired the prcticl pplictions of Yng nd Mnning s odel. Other pplicble odels bsed on discrete continuous rndo processes were proposed by Sobczyk nd Spencer [4]. Bogdnoff nd Kozin [5,6] explored the Mrkov chin theory nd utilized it to crete discrete nd continuous ftigue crck growth odels. In erlier studies, Lin proposed Fokker-Plnck eqution tht reltes the continuous Mrkov process [7]. 46
The Yng nd Mnning odel is used in reference [8] to nlyze the vrible type loding becuse of its verstile functionlity. This odel utilizes only the crck growth rte nd crck length dt; the infortion bout loding nd teril is not eployed into the odel nd is ccounted for in the rndo coponent nd odel preters. For instnce, with trnsitionl loding the odel preters will vry s the ftigue dge propgtes. The odel preter vribility ws tken into ccount in the dt driven prt of the nlyticl crck exceednce probbility, which is the probbility tht the crck length will exceed nuber of cycles, with the respective lod period. To directly ccount for the vrince in the crck growth rte, the rndo coponent is ssued to follow lognorl distribution [9,,]. A significnt prt of in pipelines re subjected to externl crcking, which is serious proble for the pipeline industry like, for exple, in Russi [], in U.S., nd in nd [3]. Identifiction of externl crcks is chieved using different ondestructive Evlution (DE) ethods. If crcks re reveled during inspection, their influence on the reining life (RUL) of the pipeline should be ssessed in order to choose wht intennce ction should be used: do nothing/repir/replce. Pipeline integrity is ssessed on the ssuption tht soe defects fter In-Line Inspection (ILI) y be: still undetected; detected, but not esured; detected nd esured. It is possible to updte the stochstic rennt life of pipelines using the dt vilble due to ILI. A robust pipeline filure odel is needed tht could be used in prctice. Usully pipelines deonstrte non-liner behvior of the teril. Becuse of this, the toughness frcture criteri is used in reference [4], described by the J-integrl of non-liner frcture echnics. The J-integrl is good descriptor of crck growth. The works of Tishev [4] describe new prcticl ethod of updting the stochstic reining life of pipelines with defects using the ltest ILI dt. It describes coprehensive lgorith for ssessing pipeline rennt life tking into ccount the results of holistic sttisticl nlysis of In-Line Inspection (ILI) dt. 47
It is ssued tht the pipeline segent wll hs longitudinl externl crck of seiellipticl for nd is described by the J-integrl. The Liit Stte Function (LSF) is described s the difference of the criticl nd current vlue of the J-integrl. The criticl crck depth is defined using the notion of frcture toughness nd the J-integrl pproch. IV.. - Definition of the J-Integrl onsider nonliner elstic body contining crck (figure 4.). Y rck O X ds Figure 4. - onliner elstic body with crck. n The J-integrl is defined s: J Γ wdy T i ui ds x () Where w ε ij ij d ε is the strin energy density, ij T n is the trction vector, Γ is n rbitrry contour round the tip of the crck, n is the unit vector norl to Γ ; re the stress, strin, nd displceent field, respectively. i ij j, ε, nd u The defined J-integrl is pth-independent line integrl nd it represents the strin energy relese rte of nonliner elstic terils: J dπ da () Where Π U W is the potentil energy, the strin energy U stored in the body inus the work W done by externl forces nd A is the crck re. 48
The probbility of filure ssessent lgorith is bsed on the Adptive Iportnt Spling (AIS) procedure. Finlly, the results of the ltest ILI re fused into the lgorith, providing best possible ssessent of pipeline rennt life s rndo vrible. The reining life updte for pipeline segent with crck-like defects using ILI dt tkes into ccount three possible outcoes: defect not discovered: defect is discovered but not esured; defect is discovered nd esured. This result perits solving ost iportnt probles of pipeline intennce: prioritiztion of pipeline segents for repir/rehbilittion; optiiztion of the tie between ILI; iniiztion of pipe opertionl risk. Model-bsed prognostic techniques rely on dynic odel of the predicted process. This pproch uses theticl odel of the process in order to ipleent the physicl understnding of the syste into the dignostic proble. Such odels should describe both noinl nd fulty behvior of the syste. As result, it is possible to explin the fult progress in tie, nd to ke End of Life (EOL) nd RUL predictions. These ethods involve the estition of residuls s devition between the rel syste esureents nd proposed odel outputs. In the idel cse, the residuls re zero but in relity there re pernent noise nd odeling errors. It is, therefore, expected tht the residuls re sll in the noinl working ode nd lrger in the presence of filure. Once the residuls re obtined, it is possible to use soe sttistic representtion to estite the distribution of RUL s function of present uncertinties nd to clculte possible dge. The syste odeling considered by the physics-bsed prognosis is derived by using physics lws nd principles. rck initition odels ust include ll the vilble infortion bout coponent nd its environent. The crck propgtion odels cn be divided into two in groups: deterinistic nd stochstic. Deterinistic crck propgtion odels, which usully describe the growth of the crck, re bsed on Pris lw [5]. Stochstic crck propgtion involves odels with rndo preters which cn be estited using Monte rlo siultions. In relity, ll previously entioned preters re ffected by soe probbility of reliztion tht influences the resulting RUL deduced fro D(). The spling of the bsic 49
preters for lrge nuber leds to curves of D() fro which we cn copute the en curve D () nd the stndrd devition (D()). Two industril pplictions re considered in order to prove the efficiency of the proposed odel. The evlution of the lifetie of suspension dping systes is considered s the in prt of the vehicles prognostic purpose. The in source of suspension filure is the ftigue occurrence due to the rod profile fluctutions. The life prognostic of petrocheicl pipelines is vitl in their doin since their vilbility hs crucil consequences. Ftigue filure is their in filure cuse due to internl pressure-depression vrition long tie. Usully, three situtions for these pipes exist: unburied, buried nd under se wter (offshore pipes). Ech one of these situtions requires different physicl preters like: corrosion, soil pressure nd friction, wter nd tosphere pressure. Hence, in the present chpter, the two in pplictions re treted s follows: First of ll, the prognostic study is pplied to predict the lifetie of suspension syste for the cses of liner nd nonliner dge ccuultion in stochstic condition where one nd two rndo vribles re considered nd which re the initil crck length nd the rod profile. Secondly, the prognostic study is pplied to buried, unburied, nd offshore pipes tking into ccount the liner nd nonliner dge cses nd considering one nd two rndo vribles which re the initil crck length with lognorl siultion nd the internl pressure P with tringulr siultion bsed on three odels: uniforly spling of the instnt T, one-tringulr period, nd ulti-tringulr period. IV.3 - Stochstic Liner Dge Accuultion To estite the residul lifetie in ftigue filure risk, n nlyticl prognostic odel presented in hpter II [6,7] is giving RUL prediction tool, whenever nlyticl physicl lws exist. Such physicl lws re: Pris-Erdogn [5] nd the liner dge ccuultion of Plgren-Miner [8] lws. 5 The nlyticl prognostic odel consists of the evlution of norlized degrdtion indictor D ( D ) in ters of lod cycle nuber. The ftigue filure is
reched when the crck size grows to criticl size with respect to Pris' lw where the necessry nuber of cycles is the criticl nuber. Using Miner cuultive dge, fter ech one lod cycle, the dge indictor D increses by reltive crck length increent d s indicted by the following eqution: D d j j (3) In hpter III, n enhnceent ws de on the nlyticl odel in order to introduce the nonliner spect of the dge ccuultion []. This enhnceent using nonliner dge function D() llows to perfor ore ccurte prognostic evlution. d d The deterinistic Pris' lw is given by the following forul: [ ] nd K( ) Y ( ) π ; K( ) Where, is the crck length, is the lod cycle, nd re the teril nd environent preters ( < <<) ; ( 4) [9], K() is the stress intensity fctor rnge, Y() is the geoetric fctor function of the body diensions, is the pplied stress rnge. IV.4 - Stochstic Modeling The stochstic odeling [3,] is considering soe influent preters s rndo vribles nd hence, the Pris' lw becoes stochstic crck propgtion lw. The dignostic dt perit to consider the initil crck length s the in rndo vrible where the second vrible is the stress loding. Mny other preters cn be lso considered s rndo nd the stochstic prognostic odel cn be expressed by the following generl function: ~ D( ) Prog ( ) ~ fct ( ~, loding ~, thickness e~, diensions,, ~,...) 5
The degrdtion indictor D vrint fro to gives us instntneously the reining useful lifetie (RUL) in ters of tie, or cycle, or distnce, depending on the type of the concerned device. ~ A probbiliztion of the bsic preters leds to probbilistic trjectory D ( ). Therefore, bundle of curves D() is obtined for which en vlue nd stndrd devition cn be deduced. Hence, chrcteristic curve D K () cn be coputed in ters of frctl α% tht depends on the level of the cceptble risk. The chrcteristic RUL is then deduced fro D K (). All previously entioned bsic preters re ffected by soe probbility of ~ reliztion tht influences the resulting RUL deduced fro D ( ). ontrry to the deterinistic-bsed prognosis, the RULs concluded in stochstic-bsed prognosis re relted to the probbilistic spect. These relevnt bsic preters ust be odeled stochsticlly using convenient well known probbility distribution lws. For exple, the initil crck length cn be odeled by either norl or lognorl distributions, the loding is odeled by norl distribution. IV.5 - Stochstic RUL The lst preters ust be odeled stochsticlly using convenient probbility distribution lws. When this is not tken into considertion, the prognostic results y not reflect relly the evluted lifetie of device. The estited RUL is then no longer deterinistic, but ffected by soe risk percentge in order to be relized. Hence bundle of RULs trjectories cn be plotted. Knowing tht the RUL cn be expressed by vrious fors like for exple in ftigue by: crck length, or criticl nuber of cycles, or teril tencity K I depending on the chosen liit sttes: service liit stte ( ), or lifetie liit stte ( ), or strength liit stte ( K 5 KI ).The RUL dopted in this work is the lifetie liit stte: - which is expressed in ters of the nuber of loding cycles.
IV.6 - Relibility Evlution of Dge Stte Ech of the liit sttes cited bove is function of rndo vribles tht kes the lso rndo functions in their turn. For this reson, they occur with certin probbility. The evlution of these probbilities is the in gol of this section. This cn be done by ny relibility ethods. The ter relibility is the probbilistic evlution of liit stte perfornce on doin of bsic vribles. In other words, it is obtined by the coputtion of the filure probbility towrd criterion or liit stte. The ethodology is s follows: ) Identify the liit sttes tht govern the lifetie of the structure. ) Identify the bsic preters intervening in these liit sttes. 3) Deduce their probbility density functions. 4) opute the filure probbility tht quntifies the risk of non-stisfction of these liit sttes. Mny types of ethods exist: the Monte rlo siultion, the pproxite ethod FORM (First Order Relibility Method), nd SORM (Second Order Relibility Method). The Monte rlo siultion ethod is bsed on lrge nuber of siultions, it is tie consuing tool nd we ust use siultions when we wnt to evlute probbility of order of - (+4) (i.e. for very sll probbility of filure, huge siultion nuber is needed). The pproxite ethod FORM is n itertive procedure tht llows clculting n index of relibility (denoted β). The index β is the distnce between the origin nd the liit stte eqution G(t) in stndrd spce. Once we hve clculted β we cn deduce the filure probbility: P rob Φ( β ) 53
In FORM pproxition the rel liit stte (usully nonliner) is replced by its tngent plne t specific point clled the ost probble filure point (MPFP). This point is the closest point on the curve: G(t) fro the origin. The liit stte G(t) divides the spce into two regions: First region where G(t) > clled sfe region. And the second region where G(t) clled filure region. Other ethods i to evlute the probbility of success of perfornce by ens of the reconstruction of the syste response PDF (probbility density function) under n nlytic for. In SORM pproxition the rel liit stte (usully nonliner) is replced by its tngent prbol t the point MPFP which is the closest point on G(t) to the origin. The liit sttes re the functions of perfornce or of stisfction of soe criteri. In our odel we re interested in the criteri of lifetie; in ftigue cse the servicebility liit stte is usully used. The servicebility liit stte governs the crck length () t cycle, in order to be under the llowble liit. This function is given by: G ( ) (4) The probbility of filure is: P rob ( G ) P ( ) f ( ) d rob (5) The probbility of success is: P rob rob f ( G > ) P ( < ) ( ) d (6) With: Y KI ( ) π x 54
IV.7 - Stochstic Bsic Preters IV.7. - Initil rck Width ƒ () The esureents of the initil crck length derived fro sensors output re treted s reliztions of rndo vrible ~. Here we consider Probbility Density Function (PDF) for tht follows lognorl distribution (figure 4.), then: Figure 4. - PDF of the crck length. f ( ) exp ) λ ξ π ξ ( Ln( ) (7) With: ξ is the stndrd devition of the vrible Ln( ) which is the equivlent norl distribution, λ is the en of the vrible Ln( ), Expecttion of : E ( ) exp[ λ + / ], ξ Vrince of : V ( ) exp[λ + ξ ] ( exp[ ξ ] ) Inversely, we hve lso: λ Ln [ E( )] V ( + ) ξ Ln E( ) V ( + ) Ln E( ) ξ V ( + ) Ln E( ) (8) (9) The llowble vlue of the crck length ( ) is fixed when the nuber of cycles reches the criticl vlue ( ) (figure 4.3). The probbility of ftigue filure is given by: Where f ( ) is the PDF of the crck width t cycle. P rob ( > ) f ( ) d 55
It cn be ssued tht e / 8 [9], where e is the device diension in the crck direction (figure 4.4). () Frcture rck length Mcrocrck initition Miniu detectble crck length Figure 4.3 - Pre-crck ftigue dge. e rck length (t) Syste stte (t) Filure Filure level Filure Stochstic degrdtion Tie t t t t t t + t + t Figure 4.4 - Probbilistic crck growth. IV.7. - PDF of rck Length t Loding ycle Since the initil crck length is rndo vrible, it is expected tht the crck length t cycle is lso rndo nd is denoted by ~. 56
57 To clculte the PDF of ~, we proceed s follows: Fro Pris' lw we cn deduce: [ ] [ ] d K d K d d ) ( ) ( Where, If we integrte the two sides between the initil stte nd n rbitrry stte, we get: [ ] [ ] [ ] [ ] d Y Y d Y d K d d K d / / / / ) ( ) ( ) ( ) ( ) ( π π π [ ] ( ) ) where ( ; / ) ( / ) (. ) ( / / / / / / / Y Y d Y π π π ( ) ( ) ) / /( / / / / / / / / / ) ( ) ( / Y Y Y Y + + π π π π ( ) ( ) the body diensions. the geoetric fctor function of is Where nd / / Y Y Y + π π Then, we hve the crck length given by the following expression: Y K π ) ( ) ( ()
58 ( ) Y + / π And the initil crck length is given by the following expression: ( ) Y / π As:, Therefore, if we hve the PDF of : ) ( f, then we cn deduce the PDF of : ) ( f, nd of : ) ( f, s follows: ) ( ) ( ( ) Jcobin Jcobin f f f Initil stte ( ) Arbitrry stte () riticl (finl) stte ( ) We cn write the following probbilistic trnsfortion: ( ) ( ) β β β β β π nd nd Let ) ( nd with the Jcobin, ) ( ) ( ) ( / Y f f d d J d d f J f f ( ) β β β π β ) ( ) ( Y f f [ ] β β ) ( A f f ( ) ) ( Where β π β Y A Then the Jcobin J is clculted s follows: [ ] ( ) ( ) ( ) ( ) [ ] ( ) β β β β β β β β β β β β β β β β β β π β β π β β π β π β β π β ) ( ) ( ) ( ) ( ) ( ) ( Y Y Y d d Y Y d d d f d d d () ()
59 [ ] β β β β A d d Since d d f f ) ( ) ( Then the PDF of is given s follows: [ ] [ ] β β β β β β ) ( ) ( A A f f IV.7.3 - PDF of the Initil Dge D We hve the reltion between the initil crck length nd the initil dge D s follows: D D D + The probbilistic trnsfortion theory gives:, As: If the proposed lw for is lognorl, then the lw of D is lso lognorl with the following PDF function: ( ) ) ( ) ( exp ) ( D Ln D f + λ ξ π ξ ; 8 nd As e D D + Then we cn write the PDF s follows: ) ( ) 8( exp ) ( D D D D e Ln D f + + λ ξ π ξ After tht we hve deterined the PDF of which is ) ( f (eqution 3), we cn clculte the probbility of filure by the following servicebility criterion: <. ) ( + D dd d ) ( ) ( ) ( D f D f + ) ( ) ( dd d f D f (3) (4)
IV.8 - Eqution of the Stochstic-Bsed Prognostic The stress rnge in ftigue is governed by the WÖhler's curve (figure 4.5). The trnsversl crck is criticl when it is norl to the stress loding rnge (figure 4.6). Applied stresses : endurnce liit e 3 3 Figure 4.5 - WÖhler's curve of ftigue. Figure 4.6 - riticl crck length perpendiculr to stress loding. The degrdtion evolution in ters of the bsic vribles is the following stochstic recursive reltion [3]: Where ~ ~ ~ D ( ~ ) D ( ~ ) + dd d D ~ is the probbilized dge increent t the end of ech loding cycle. ( ~ ) (5) IV.8. - Developent of d D ~ We hve fro Pris' lw: d d [ K( )] d [ K( )] d ; s K( ) Y ( ) j π d [ Y ( ) π ] j d As: d dd ( ) nd for d (t the end of ech one cycle) dd ( ) [ Y ( ) π ] j dd ) ( Y ( π ) j 6
For stochstic initil crck length ~, the probbilized dge increent is given under the following stochstic for: ~ dd ( ~ ) ~ ( Y ( ~ ) π ~ ~ ) j (6) Where it is ssued tht: e/8 nd. e/. Fro equtions (5) nd (6), the prognostic odel under the liner stochstic condition cn be written s follows: ~ D ( ~ ) ~ ( Y ( ~ ) π ~ ~ ) D ) + dd ) D ) + ~ ( ~ ~ ( ~ ~ ( ~ j (7) The previous reltion describes the degrdtion evolution in ters of the following rndo vribles: initil crck size ~, loding ~ j, nd the current crck size ~. This reltion represents the stochstic recursive prognostic odel s it perits to relte the degrdtion indictor D ~ to the bsic rndo vribles. At ech loding cycle ( ), the degrdtion indictor D increents of quntity dd strting fro D till reching the unit vlue (D ) which is the filure stte. Eqution (7) gives the reliztion of the stochstic degrdtion t cycle. The preters nd re the vribles with the environent nd the teril properties, these preters cn lso be tken s rndo vribles. IV.8. - Developent of d ~ Inversely, in ters of crck width, the degrdtion cn be expressed by the crck length increent t the end of ech one loding cycle (d ) by the following recursive reltion: d~ ~ ~ ( Y ( ~ ) π ~ ~ j ) + d~ ~ + Y ( ~ ) ( π ~ ~ ) In the following sections, we will pply the proposed prognostic odel (equtions 7 nd 8) to industril systes like vehicle suspensions nd petrocheicl pipelines. j (8) 6
IV.9 - Flowchrt of the Stochstic-Bsed Liner Prognostic Dignostic/Inspection Input initil preters (e,, ); Sensor esureents ( ) nd geoetric function (Y()) Estition of criticl crck length e/8 Stochstic odeling of : f ~ ( ) For ech lod cycle, Anlytic siultion of crck growth Stochstic odeling of : d f ~ ( ) f ~ ( ) d Lod siultion: ~ j Stochstic crck length: d~ Y ( ~ ) π ~ ~ j ~ ~ d~ + ( ) Degrdtion liner ccuultion Stochstic degrdtion: ~ ( ~ dd ( ~ ~ ~ ) ~ Y ) π j ~ ~ ~ D ( ~ ) D ( ~ ) + dd ( ~ ) ( ) D ~ ( ~ ) < Yes o Record: criticl cycle RUL t cycle : RUL() - Plot ( ) ( D ; ), Prognostic ( ) Plot ( RUL( ) ; ), 6
IV. - Appliction to the Suspension Syste Referring to hpter II, the se utootive suspension syste is tken in this section s n industril ppliction. Two kinds of preters re present in this ppliction, deterinistic preters nd rndo preters. The two rndo vribles in this ppliction re the initil crck length nd the rod profile vrition x tht cretes rnge of stresses. onsider the sttisticl lognorl preters of the initil crck length which re presented s follows: Men vlue (or expecttion): E ( ). Stndrd Devition nd Vrince: ( 6 ).945 V ( ) ( ) (.945) 8.673 And the sttisticl preters of the initil dge D cn be deduced s follows: E( ). Men vlue (or expecttion): E( D ) E. 8 c c E( ) /8. Vrince: V ( D ) V c. /8. E( ) E( c 8.673. 6 + V ( ) V ( c ) OV (,( c +. ) E( ) E( c ) E( ). E( c 8.673 6 ( /8.) OV (,(. E( ). E( c c )) ) )) ) ote : OV ( X, Y ) E( X. Y ) E( X ). E( Y ) E( [ E( )] V ( ) + [ E( )] ) OV (, ) + OV ( c 6,( )) V ( ) 8.673. V ( D ) /8..65 ( /8.) 4 8 6 [.685 +.4 + 3.497 ] 7 4 ( D ) V ( D ).43 3.784 8.673. 6 + 6 6 8.673 8.673 +...(/8.) 7.43 Moreover, the equivlent norl preters of re deduced s follows: 63
V ( ) 8.673 λ Ln[ E( ) ] Ln (.).5 + ( ) Ln Ln + E.4 4.694.5.68.695 V ( ) ξ Ln + ( ) ξ E V ( ) Ln + ( ) E.68 4 8.673 Ln +.4.474 6 6 The stress rnge in the suspension in ters of the rod profile rnge is siplified by the following expression: Where, j j E x (9) ll : is the length of the suspension device (ll 5 ) x j : is the vrition of this length (diltion) under profile excittion (see tble 4.). E: is the Young's odulus of the suspension teril (E GP). We study two cses: the cse of one rndo vrible ( ~ x j ) rndo vribles which re the ( ~ ) nd the initil dge ( ~ ). x j nd the cse of two IV.. - Liner Stochstic se This cse is treted for one rndo vrible nd two rndo vribles. IV... - One Rndo Vrible We consider here the cse of liner dge (Miner's lw) with one stochstic ~ norlly distributed fro which we deduce the preters of the pplied preter ( x j ) stress rnge ( ~ j ) s follows: ~ : orl Lw E E( ~ j ) x E V ( ~ j ) V j ( ~ x ) j (for ech ode of rod profile) The sttisticl preters for ech ode of rod profile re surized in tble 4. below. 64
Tble 4. - Sttisticl chrcteristics of ech ode of rods profile. Rod Mode Severe (ode ) Fir (ode ) Good (ode 3) Men of ~ x ( x j j in ) oefficient of Vrition of ~ (in %) x j Stndrd Devition ~ (in ) ( ) x j Lw 5% 5 orl 5 % 5 orl 5 5%.5 orl Fro the siultion of the stochstic prognostic odel proposed under eqution (7), the degrdtions evolution of the suspension is obtined nd presented in figure 4.7 below. Figure 4.7 - Suspension degrdtion under liner dge lw nd stochstic rod excittions. The lifeties noted fro figure 4.7 re s follows: Mode :,, cycles. Mode : 3,995, cycles. Mode 3: 6,9,5 cycles. IV... - onversion of Lifeties into Yers To convert the suspension lifetie into yers' unit, ssue tht new rod profile reliztion occurs ech seconds. If we ssue lso tht the suspension tie usge is % of 65
dy (.4 hours/dy) then the expected lifeties' durtions re (refer to hpter II, Prgrph 3..):,,(cycles) (s) For ode :.64 yers 6(s) 6(in).4(hours) 365(dys) For ode : 3,995,(cycles) (s) 6(s) 6(in).4(hours) 365(dys).53 yers For ode 3 : 6,9,(cycles) (s). yers 6(s) 6(in).4(hours) 365(dys) IV... - Two Rndo Vribles In this section, two stochstic preters re considered for the liner cse of dge ccuultion nd which re the following: ) The rod excittion effect : ~ : orl Lw ~ E E( j ) x ( ~ E V j ) V j ( ~ x ) j (for ech ode of rod profile) ) The initil crck length : ~ : Lognorl Lw E( ~ ). ( ~ 6 V ) 8.673 ( ~ ) V ( ~ ).945 The preters of the rod profiles ( ~ ) re given in tble 4.. x j The results of degrdtions evolution of the suspension re presented in figures 4.8 nd 4.9 below. 66
Figure 4.8 - Suspension degrdtion under liner lw of dge nd stochstic rod excittions nd initil crck width..46.79.535.455.785.53.45.78.55.445.775.5 D.44 D.77 D.55.435.765.5.43.76.55.45.755.5.4.5 3 3.5 4 4.5 5 5.5 6 uber of cycles x 5.75.3.35.4.45.5.55.6.65 uber of cycles x 6.495 7.5 7.55 7.6 7.65 7.7 7.75 7.8 7.85 uber of cycles x 6 Figure 4.9 - Zoo-in for the three cses. Fro the zooing-in shown in figure 4.9, we note tht the fluctutions of the curve due to stochstic effects increse s the rod condition gets better. This phenoenon cn be explined by the fct tht the stochstic dispersion preters re ore influent in good rod condition (ode 3) cse thn in severe condition (ode ) where the en rod profile is uch higher (tble 4.). By coprison to the cse of one rndo vrible it is cler tht the lifeties decrese for the three odes s follows: 67
One rndo vrible Two rndo vribles Decrese (%) Mode,, cycles 6, cycles 39.6% Mode 3,995, cycles,7,5 cycles 3.% Mode 3 6,9,5 cycles,5, cycles 36.9% The conclusion drwn here is it is iportnt to consider ll preters s rndo when these preters show soe sensibility on the lifetie vlue. IV... - onversion of Lifeties into Yers To convert the suspension lifetie into yers' unit, ssue tht new rod profile reliztion occurs ech seconds. If we ssue lso tht the suspension tie usge is % of dy (.4 hours/dy), then the expected lifeties' durtions re (refer to hpter II, Prgrph 3..): 6,(cycles) (s) For ode :.39 yers 6(s) 6(in).4(hours) 365(dys) For ode : For ode 3 :,7,5(cycles) (s).7 yers 6(s) 6(in).4(hours) 365(dys),5,(cycles) (s) 6.44 yers 6(s) 6(in).4(hours) 365(dys) IV... - oprison: Deterinistic - Stochstic Results (for Liner Dge Lw) Deterinistic cse Stochstic cse ( RV) 68 Figure 4. - Deterinistic nd stochstic study of suspension degrdtion under liner dge lw.
Fro figure 4., it cn be noted tht in the stochstic cse the lifeties re reduced significntly reltively to the deterinistic cse like s follows: Mode (severe condition) : fro 6,836, cycles to 6, cycles (nerly 9.%) Mode (fir condition) : fro,85, cycles to,7,5 cycles (nerly 75.%) Mode 3 (good condition) : fro 7,, cycles to,5, cycles (nerly 4.%) It is logicl conclusion since the stochstic effects re generlly negtive on the suspension lifeties. In fct, it is known tht the dispersions (stndrd devitions) introduced by these rndo vribles (lod stresses induced by rod profile nd initil crck length of suspension) propgte through ll the degrdtion equtions nd resulting in reduced lifetie vlues. Moreover, the better the rod conditions the sller the lifetie reductions. IV...3 - RUL Evlution of Suspension in Stochstic se The globl RUL evlutions re deduced fro the expression -. In fct is the necessry cycle nuber to rech filure (ppernce of the first cro-crcks) nd is the initil cycle nuber t the beginning of service tken generlly equl to. These curves decrese fro totl lifetie of the device to zero where D D. Fro these curves we cn deduce t ech instnt the reining useful lifetie of the device (RUL - ) nd hence, the prognostic result cn be inferred (figure 4.). RUL 6, cycles RUL,7,5 cycles RUL 3,5, cycles Figure 4. - RUL evolution of the suspension stochstic degrdtion under liner dge lw. 69
IV...3 - Vlidtion of the Suspension Life under Liner Dge Rule The vlidtion of these results cn be found in the work of reference [3] on the ftigue life of suspensions. An verge life of,375 k is deduced under norl conditions nd which corresponds to.3 yers for vehicle running with 5 k per hour nd for.4 hours per dy. IV.. - onliner Stochstic se IV... - Stochstic onliner uultive Dge The cse of ftigue degrdtion tken in the precedent section is theticlly forulted nd bsed on the nlytic lws of Pris nd Miner. The lst lw is liner cuultive dge odel. Its lrgest drwbck is its inbility to ccount for the order of loding. Tht is, the resulting filure prediction is independent of the lod interction effects tht hve been observed between high-cycle nd low-cycle lodings. Pst reserch hs shown there is nonliner interction effect between high cycle ftigue (HF) nd low cycle ftigue (LF) in ny engineering terils. This effect hs been observed within unixil lodings, but is often ore pronounced under ultixil loding, prticulrly when the loding is non-proportionl. The nonliner interction effect precludes the use of the liner dge rule for dge ccuultion. In the present study, the effect of HF loding hs hd ore detrientl effect when coupled with the LF lodings thn predicted by liner sution rule. onliner dge ccuultion theories cn ccount for this influence nd hve shown n iproveent in prediction. The stress levels were chosen to correspond to levels previously tested to filure, resulting in ftigue lives rnging fro pproxitely 5 to 7 cycles. A nonliner dge sution is required to properly define the ftigue process since the liner sution of dge given by Miner's su is often not dequte to predict the service life of coponent when subjected to vrible-plitude lodings. cycle by: The nonliner cuultive dge is deonstrted in hpter III nd given t ech 7
7 ( ) ( ) / ) ( + + + α α α D D The growth of D() t the end of ech cycle in ters of the crck width () is given by the following reltion: Where, / : is the stress plitude in one cycle, this preter is generted s n input lod whose en is tken to be equl to 8 MP, the ftigue liit (is the endurnce liit stress of teril) tken to be equl to 8 MP. Here two cses re considered: one rndo vrible (loding ) nd two rndo vribles (loding nd initil crck width ). The stochstic nonliner prognostic odel cn be written s follows: ( ) ( ) ~ ~ ~ / ~ ~ ) ( ~ D D D j + + + α α α ) ( ) ( D ~ ~ ~ () () ()
IV... - Flowchrt of the Stochstic-Bsed onliner Prognostic Dignostic/Inspection Input initil preters (e, α,,, ; ) ; Sensor esureents ( ) Estition of criticl crck length e/8 For ech lod cycle, Stochstic siultion of nd Lod siultion: ~ j Initil crck siultion: ~ Initil stochstic dge: ~ ~ ~ D ( ) ~ onliner degrdtion ccuultion ~ D( ) Stochstic nonliner degrdtion: ~ / j ~ ( ( ~ α + )) D ( α + ) α + D ~ ( ) < Yes o Record: criticl cycle RUL t cycle : RUL() - ( ) Plot ( D ( ); ), Prognostic curve Plot ( RUL( ) ; ), ( ) 7
IV...3 - One Rndo Vrible We consider here the cse of nonliner dge with one stochstic preter ~ following the norl lw (tble 4.). Fro the siultion of the stochstic prognostic odel proposed under eqution (), the degrdtions evolution of the suspension is obtined nd presented in figure 4.. Figure 4. - Suspension degrdtion under nonliner dge lw nd stochstic rod excittions. The lifeties noted fro the figure 4. re for ech ode s follows: Mode : 8,5,35 cycles. Mode :,34,9 cycles. Mode 3: 6,78, cycles. IV...3. - onversion of Lifeties into Yers To convert the suspension lifetie into yers' unit, ssue tht new rod profile reliztion occurs ech seconds. If we ssue lso tht the suspension tie usge is % of 73
dy (.4 hours/dy), then the expected lifeties' durtions re (refer to hpter II, Prgrph 3..): 8,5,35(cycles) (s) For ode : 5.4 yers 6(s) 6(in).4(hours) 365(dys) For ode :,34,9(cycles) (s) 7.6 yers 6(s) 6(in).4(hours) 365(dys) For ode 3 : 6,78,(cycles) (s).64 yers 6(s) 6(in).4(hours) 365(dys) IV...4 - Two Rndo Vribles We consider here the cse of nonliner dge with two stochstic preters: the loding fro the rod excittion ~ nd the initil crck length ~ ) The rod excittion effect : ~ : orl Lw ~ E E( j ) x ( ~ E V j ) V j ( ~ x ) j (for ech ode of rod profile) ) The initil crck length : ~ : Lognorl Lw E( ~ ). ( ~ 6 V ) 8.673 ( ~ ) V ( ~ ).945 The results of degrdtions evolution of the suspension re presented in figures 4.3 nd 4.4 below. 74
Figure 4.3 - Suspension degrdtion under nonliner lw nd two rndo vribles: stochstic rod excittions nd initil dge. Degrdtion D.35.3.5..5..5...3.4.5.6.7 uber of cycles x 5 Figure 4.4 - Zoo in for the three cses. The lifeties noted fro the figure 4.3 re for ech ode s follows: Mode : 8,63,85 cycles. Mode :,69,65 cycles. Mode 3: 6,88, cycles. Fro the zooing-in shown in figure 4.4, we note tht the fluctutions of the curves due to stochstic effects re siilr for ll rod conditions. This cn be explined by the fct tht in nonliner dge, the stochstic dispersion effects dointe for ll rod conditions. indicted: By coprison to the cse of one rndo vrible, the following lifeties re 75
One rndo vrible Two rndo vribles Increse (%) Mode 8,5,35 cycles 8,63,85 cycles.% Mode,34,9 cycles,69,65 cycles.% Mode 3 6,78, cycles 6,88, cycles.6% ontrrily to the liner cse, the lifeties increse fro one rndo vrible to two rndo vribles for ll odes; this conclusion is explined by the fct tht the nonlinerity dointes the stochstic effect. IV...4. - onversion of Lifeties into Yers To convert the suspension lifetie into yers' unit, ssue tht new rod profile reliztion occurs ech seconds. If we ssue lso tht the suspension tie usge is % of dy (.4 hours/dy), then the expected lifeties' durtions re (refer to hpter II, Prgrph 3..): 8,63,85(cycles) (s) For ode : 5.46 yers 6(s) 6(in).4(hours) 365(dys) For ode :,69,65(cycles) (s) 7.5 yers 6(s) 6(in).4(hours) 365(dys) For ode 3 : 6,88,(cycles) (s).7 yers 6(s) 6(in).4(hours) 365(dys) IV...4. - oprison: Deterinistic - Stochstic Results (onliner Dge Lw) To show the stochstic effects, coprison is done between the deterinistic results nd the stochstic results (two rndo vribles cse). 76
Deterinistic cse Stochstic cse ( RV) Figure 4.5 - Deterinistic nd stochstic study of the suspension degrdtion under nonliner dge lw. Fro figure 4.5, it cn be noted tht the lifeties re reduced fro the deterinistic cse to the stochstic cse s follows: Mode (severe condition): fro 9,47,7 cycles to 8,63,85 cycles (nerly 4.8%) Mode (fir condition) : fro,63,8 cycles to,69,65 cycles (nerly 6.6%) Mode 3 (good condition) : fro 8,95,4 cycles to 6,88, cycles (nerly 6.7%) It is noted tht ore the rod conditions becoe better ore the lifetie reductions becoe greter. Moreover, the fluctutions in stochstic curves re due to the stochstic dispersions (stndrd devitions). In fct, the stochstic effects re generlly considerble on the suspension lifeties due to the dispersions introduced by these rndo vribles tht propgte through ll the degrdtion equtions nd resulting in reduced lifetie vlues. The finl rerk is tht the stochstic effects dointe here over the nonliner effects in lifeties estitions. Hence, it is iportnt to include the stochstic effects for ore relistic prognosis under the condition tht we consider relible sttisticl dt for the initil crck widths nd the rod profile excittions. 77
IV...5 - Vlidtion of the Suspension Life under onliner Dge Rule The vlidtion of these results cn be found in the work of reference [3] on the ftigue life of suspensions. An verge life of 3, k is deduced under norl conditions nd which corresponds to 7.35 yers for vehicle running with 5 k per hour nd for.4 hours per dy. IV. - Appliction to the Pipeline Systes to Three ses We restudy the prognostic of the pipeline syste lredy treted in hpter II; this, by tking into ccount the liner nd the nonliner dge lw but this tie for the stochstic cse of vribles [33]. The study is done for one nd two rndo vribles (internl pressure P nd initil crck length ). The geoetric properties of pipes re presented in hpter II. Three xil levels of internl pressure P re considered (tble 4.) with repetition period T P. At ech of these levels, degrdtion trjectory D() is deduced in ters of cycle nuber. When D() reches the unit vlue, then the corresponding is the lifetie of the pipe tht filed by ftigue. We siulte three odes of P j with the sttisticl preters given in tble 4.. Tble 4. - The three pressure odes. Pressure P (MP) Mode High (ode ) 8 Middle (ode ) 5 Low (ode 3) 3 IV.. - Eqution of the Stochstic-Bsed Prognostic In the cse of pipes of thickness e, the stress rnges re creted by the pplied internl pressure; hence, the following reltion gives the criticl hoop stress rnge pressure rnge P (figure 4.6): θ in ters of the θ L P R e (3) 78
P(t) Internl Pressure P P t T T T Figure 4.6 - Tringulr pressure lw. The siultion of the internl pressure following tringulr lw P ~ (figure 4.6) genertes sple of stress rnges ~ following the se tringulr lw fro the eqution below: Pj R j knowing tht : Pj Pj P e j Fro the following eqution: ~ dd ~ ( Y ( ~ ) π ~ ~ ) j (4) It cn be deduced tht: ~ dd ~ ( e / 8 ).6 ~ + ( / e) 3 ( ~ / e) π ~ P ~ j. R e (5) Where, Y ( ).6 + ( / e) 3 ( / e) is the geoetric function of the pipes. IV.. - Genertion of Internl Pressure P i The Monte-rlo siultion of the rndo P i is copleted using three odels: Model A) : Tringulr with unifor spling of tie t; Model B) : Over one initil tringulr period T P ; Model ) : Over ulti tringulr periods T P. 79
IV... - Monte-rlo Siultion Principle The Monte-rlo siultion (figure 4.7) consists of rndo spling of lrge nuber of u in [,] intervl with the se probbilities (using the unifor distribution). As i ( i i i i i u F u ) (the second bisector) nd hence u F ( u ) F ( x ), x F ( u i ). The U U X X genertion of F X (x). i x leds to the reconstruction of the rndo vrible sple following the lw F U (u) F X (x) Unifor Lw U(,) Generl Lw ½ u i ½ u i i x F ( u ) Figure 4.7 - Monte-rlo siultion principle. X x Where u is the unifor-bsed generted vlue. IV... - Model A: Unifor Genertion of Tie t Here, the tringulr pressure P j is siulted t ech instnt t considering unifor distribution for the tie t [,]. P(t) Pressure (MP) P P (t) P (t) t (s) T T T Figure 4.8 - Tringulr siultion of the pressure in ters of unifor tie spling. 8
8 The siulted pressure digr is given in ters of tie t by the following function: > + ; ~ ) ( ; ~ ) ( ) ( T t if P t T P t P T t if t T P t P t P Where the vrible t is siulted rndoly under unifor lw (figure 4.8). IV...3 - Model B: One Initil Tringulr Period T P In this cse, the internl pressure P is siulted by Monte-rlo ethod using tringulr distribution over one initil period of pressure T P. The tringulr lw of the internl pressure is given by the following functions (figure 4.9): The PDF function of P: > < b P P b P c c b b P b c P c b P p f P nd ) )( ( ) ( ) )( ( ) ( ) ( The uultive Density Function (DF) of P: < < < b P b P c c b b P b c P c b P P p F P ) )( ( ) ( ) )( ( ) ( ) ( ( p) f P P b c f P (c) Figure 4.9 - Tringulr PDF function of P. (6) (7) (8)
The inverse of the DF function gives reliztion P j for P s follows: P j + u ( )( ) j b c u j θ F ( u j ) b ( u )( )( ) j b b c θ u j (9) Where, u j : the unifor-bsed generted vlue in the intervl [,], c θ, b The en vlue: The vrince: V ( P) 3 ( θ ) + c + b P, 6( θ ) 3 θ ( θ ) b ( c )( b c). 8 8 ( b ) Here, the siultion of the internl pressure is copleted long one period T P under tringulr lw distribution of en vlue P : 3 f P ( P ) ( θ ) + c + b + P + P 6( θ ) 3 3 f P ( p) c b For the se initil period T P, ech siultion gives different reliztion of the PDF; thus, new vlue for c P is given, keeping lwys nd b T P. T P T P Figure 4. - Tringulr PDF of P. P We consider the following vlues for the siultion (figure 4.): ; b T P (pressure intervl); nd c P (pressure vlue). Where the period T P is pressure intervl tht cn be tken s percentge of the xil pressure P. IV...4 - Model : Multi-Tringulr Period In this cse, we do the Monte-rlo siultion of the syetric tringulr distribution repeted stochsticlly long tie with respect to pressure period T P. In ech period, new siultion gives different reliztion of the density function; thus, new vlues for, b, c re given ech tie (figure 4.). 8
( p) f P b P c b c b c b T P T P T P T P Figure 4. - Multi-Tringulr PDF function. We tke the following vlues for ech siultion: i T P ( i instnts:,,,...) ; b +T P ; c (b+)/ IV..3 - Liner se of Dge In this prt, the liner Miner's lw of dge is used. One nd two rndo vribles re considered nd which re the pressure P nd the initil crck length. The siultion odel dopted here for pressure P is the tringulr lw in ters of unifor siultion of tie t (odel A). IV..3. - One Rndo Vrible (Pressure) As for the deterinistic study executed in hpter II, the study encopsses three odels for pressure genertion (tble 4.3) nd three types of pipes: unburied, buried nd offshore. Tble 4.3 - Sttisticl chrcteristics of ech pressure ode. Pressure P j (MP) δ Pj (%) Lw Mode High (ode ) 8 % Tringulr Middle (ode ) 5 % Tringulr Low (ode 3) 3 % Tringulr IV..3.. - Model A for Pressure Genertion For the cse of odel A pressure genertion, the degrdtion evolutions for the unburied pipes re given in figure 4.. We note here the following lifeties: 4.8 yers 83
(High pressure), 6.75 (Middle pressure), nd 9. yers (Low pressure). The results show steep increse of degrdtion fro the 4 th yer onwrd for the High ode while it is fro 6.5 yers for the Middle ode nd fro 9 yers for the Low ode. Figure 4. - Unburied pipelines under liner dge lw nd stochstic P. In buried pipes cse, the degrdtion evolutions for the cse of odel A of pressure genertion re given in figure 4.3. The following lifeties re noted: 4.5 yers (High pressure), 6.3 (Middle pressure), nd.5 yers (Low pressure). The results show lso shrp increse of degrdtion fro the 4 th yer onwrd for the High ode while it is fro the 6 th yer for the Middle ode nd the Low ode shows ore progressive increse in degrdtion with tie. Figure 4.3 - Buried pipelines under liner dge lw nd stochstic P. 84
The degrdtion evolutions for the offshore pipes for the cse of odel A of pressure genertion re given in figure 4.4. We note here the following lifeties: 8.5 yers (High pressure), yers (Middle pressure), nd 33 yers (Low pressure). The results show progressive increse of degrdtion long tie for ll pressure odes except for the Low ode where steep increse is noted fro 3 yers fter cler progressive degrdtion. Figure 4.4 - Offshore pipelines under liner dge lw nd stochstic P. IV..3.. - Model B for Pressure Genertion For the cse of odel B pressure genertion, the degrdtion evolutions show different results fro the odel A. In fct, for the unburied pipes, the results re represented in figure 4.5. We note here the following lifeties:.9 yers (High pressure), 4. yers (Middle pressure), nd 6.5 yers (Low pressure). The results show progressive increse of degrdtion for ll odes except for the High nd Middle odes where steep increses occur t the finl stge. Figure 4.5 - Degrdtion evolution for unburied pipe under stochstic P. 85
The degrdtion evolutions for the buried pipes for odel B of pressure genertion re given in figure 4.6. The following lifeties re noted: 8. yers (High pressure),.3 yers (Middle pressure), nd 5.9 yers (Low pressure). The results show progressive increse of degrdtion for ll odes especilly for the lst ode. Figure 4.6 - Degrdtion evolution for buried pipe with stochstic P. Finlly, for odel B of pressure genertion, the degrdtion evolutions for the offshore pipes re given in figure 4.7. We note here the following lifeties: 8 yers (High pressure), 6 yers (Middle pressure), nd.5 yers (Low pressure). The results show progressive increse of degrdtion for ll odes especilly for the lst ode. 86 Figure 4.7 - Degrdtion evolution for offshore pipe under stochstic P.
IV..3..3 - Model for Pressure Genertion For the cse of odel pressure genertion, the degrdtion evolutions for the unburied pipes re given in figure 4.8. We note here the following lifeties:.9 yers (High pressure), 4. yers (Middle pressure), nd 6.5 yers (Low pressure). The results show steep increse of degrdtion for the odes High, Middle, nd Low fro the yers:.5, 3.5, nd 6 respectively. Figure 4.8 - Degrdtion evolution for unburied pipe with stochstic P. The degrdtion evolutions for the buried pipes in the cse of odel of pressure genertion re given in figure 4.9. We note here the following lifeties: 8 yers (High pressure),. yers (Middle pressure), nd 7. yers (Low pressure). The results show steep increse of degrdtion for the odes High, Middle, nd Low fro the yers: 6.5,.5, nd 6.8 respectively. Figure 4.9 - Degrdtion evolution for buried pipe with stochstic P. 87
For the offshore pipes in the cse of odel of pressure genertion, the degrdtion evolutions re given in figure 4.3. We note here the following lifeties: 9 yers (High pressure), 3.5 yers (Middle pressure), nd yers (Low pressure). The results show progressive increse of degrdtion for ll odes especilly for the lst ode. Figure 4.3 - Degrdtion evolution for offshore pipe with stochstic P. IV..3. - Two Rndo Vribles: Pressure (One Tringulr Period) - (Lognorl Lw) Here, for ech instnt, the siultion of the internl pressure is done long one initil period T P (odel B) under tringulr distribution lw of en vlue P : P + c + b 3 + P + T 3 P For the se initil period, ech siultion gives different reliztion of the density function; thus, new vlue for c P is given, keeping lwys nd b T P. We consider the following vlues for the siultion: ; b T P (pressure intervl); nd c P (pressure vlue). The Tringulr siultion of the internl pressure, with respect to odel B nd for the three odes, leds to the pplied stress blocks shown in figure 4.3. This figure shows tht, for the three blocks of pplied stresses, the rndoness is clerly illustrted by the fluctution vlues of these stresses with the cycle nubers. The en vlues of the three blocks re respectively 4 MP, 5 MP, nd 9 MP. 88
5 45 4 Mode : High Applied Stress Blocks Stress rnge (MP) 35 3 5 5 Mode : Middle Mode 3 : Low 5 3 (cycles) 4 5 6 x 5 Figure 4.3 - Applied stress blocks on pipes for three odes of pressure. The initil crck length is siulted long lognorl lw with the following preters: ~ E( ~ ( ~ ). : Lognorl Lw ) ( ~ V ).945 The crck length (t) growth versus tie is given in figure 4.3 tht shows for the three odes the length evolution fro n initil vlue to the criticl vlue e/8. They grow fro n initil vlue. to the end of life where ll curves (t) rech the criticl width e/8. The High pressure ode revels the fstest width increse. The criticl crck lengths reched for ech pressure ode t the instnts re: 3.5 yers, 5.3 yers, nd 6.8 yers respectively. (t) () Figure 4.3 - rck length evolution with tie for unburied pipe with rndo P nd. 89
The siultion of the prognostic eqution (7) previously developed perits to drw, for ech level of pressure (High, Middle, nd Low), the degrdtion trjectory D in ters of tie t. The results of degrdtion trjectory siultions re shown in figure 4.33 below. Figure 4.33 - Degrdtion evolution for unburied pipe with tringulr P nd lognorl. onversely, t ech instnt t, the Reining Useful Lifetie RUL(t) t - t (figure 4.34) cn be deduced strting fro the rw stte of the pipe RUL(t ) t - t which gives the entire ge of the pipe, till reching the filure stte (D D ) where RUL(t ) t - t (See exple on figure 4.34 for Mode : High). The RULs for unburied pipes is nerly 3.6 yers for ode (High pressure), 5. yers for ode (Middle pressure), nd 6.35 yers for ode 3 (Low pressure). t t t 9 Figure 4.34 - RUL evolution for unburied pipe with tringulr P nd lognorl.
For buried pipes (figure 4.35), it is nerly 8.75 yers for ode (High pressure),.8 yers for ode (Middle pressure), nd 6.33 yers for ode 3 (Low pressure). Figure 4.35 - Degrdtion evolution for buried pipe with tringulr P nd lognorl. For offshore pipes (figure 4.36), it is nerly. yers for ode (High pressure), 3.7 yers for ode (Middle pressure), nd.43 yers for ode 3 (Low pressure). Figure 4.36 - Degrdtion evolution for offshore pipe with tringulr P nd lognorl. 9
The degrdtion indictor D evolves fro D to D where the pipe is t the end of its life nd this for ech pressure ode. The obtined lifetie vlues re verified to be in the rnge of rel lifeties ccording to the references [33,34]. As we cn notice, these curves re stochstic nd the lifeties deduced fro the re lso stochstic. Therefore, we do not hve unique vlue for the corresponding RUL(t), but new reliztion is derived fro ech siultion of D(t) nd the en vlues D( t) nd RUL( t) cn be inferred. IV..4 - onliner se In this cse, we dopt the nonliner lw for dge ccuultion developed in hpter III. As in the previous liner cse, we ke the stochstic study for one nd two rndo vribles. IV..4. - One Rndo Vrible (Pressure) Here, the internl pressure is the result of tringulr siultion using the odel B. Three pressure odes re considered: High (in red), Middle (in blue), nd Low (in green) where the vlues re given in tble 4.3. The results re represented by the following figures. Figure 4.37 - Degrdtion evolution of unburied pipes under stochstic P nd nonliner dge. We note fro the previous figure 4.37 tht the RULs re respectively: 3.53 yers for ode, 5.89 yers for ode, nd.6 yers for ode 3. It cn be seen clerly tht the 9
soothness of ll the curves cn be explined by the doinnce of the nonliner effect on the stochstic one. The degrdtions increse lrgely t the finl stge of their lives. Figure 4.38 - Degrdtion evolution of buried pipe under stochstic P nd nonliner dge. Fro the previous figure 4.38, it is noted tht the RULs re respectively: 8.8 yers for ode, 4.7 yers for ode, nd 6.5 for ode 3. It cn be seen clerly tht the soothness of ll the curves cn be explined by the doinnce of the nonliner effect on the stochstic one. The degrdtions increse considerbly t the finl stge of their lives. Figure 4.39 - Degrdtion evolution of offshore pipe under stochstic P nd nonliner dge. 93
Figure 4.39 shows tht the RULs re respectively:. yers for ode, 9.4 yers for ode, nd 34. for ode 3. As in the three precedent siultions, it cn be seen clerly tht the soothness of ll the curves cn be explined by the fct tht the dispersion introduced fro the stochstic condition is not very influent. Actulity, the nonliner effect here dointes the stochstic one relted to the rndo vrible P. Moreover, degrdtions increse significntly t the finl stge of their lives. The vlue obtined for pipes lifeties re logicl knowing tht the end of life does not en necessrily the totl replceent of the pipe but tht ens tht the pipe intennce should be done now. IV..4. - Two Rndo Vribles (Pressure nd Initil rck Length) In this section, two rndo vribles re considered: the pressure nd the initil crck length. We execute tringulr siultion of internl pressure P using odel B for the three odes: High, Middle, nd Low (tble 4.3). The initil crck length is siulted s lognorl distribution using the following preters: ~ E( ~ ( ~ ). : Lognorl Lw ) ( ~ V ).945 λ Ln The equivlent norl preters for re inferred s follows: [ E( )].695 V ( ) ξ Ln + ( ) ξ E V ( ) 8.673 Ln (.).5 + ( ) Ln Ln + E.4 V ( ) Ln + ( ) E 8.673 Ln +.4 6 6.694.5.68.68 4 4.474 The initil dge D is deduced fro s follows: ~ ~ D D ~ ~ D : Lognorl ~ E( D ).8 Lw ~ ~ ( D ) V ( D ) 3.784 4 94
by: The nonliner cuultive dge, previously deonstrted, is given t ech cycle ~ D ( ) ~ α + ( ) D ( α + ) ~ / j ~ D ~ ~ α + Figure 4.4 below revels the crck width growth s function of tie t. It is noted tht the low pressure ode revels the lowest increse rte (slope) in crck width in coprison with the two other pressure odes. onsequently, these two previous odes rech erlier the criticl width. (t) () Figure 4.4 - rck width evolution with tie of unburied pipe under stochstic pressure nd initil crck length for nonliner dge. The siultion of the prognostic eqution () perits to drw the degrdtion trjectory for ech level of pressure: High (red), Middle (blue), nd Low (green), by considering the three cses of pipelines. 95
Figure 4.4 - Degrdtion evolution of unburied pipe under stochstic P nd for nonliner dge. The results for unburied pipes (figures 4.4 & 4.4) show tht the pipe lifetie for this cse is nerly 3. yers for ode (High pressure), 5 yers for ode (Middle pressure), nd 9 yers for ode 3 (Low pressure). The degrdtion curves show ore steep evolution for the two first odes thn the third ode. 96 Figure 4.4 - RUL evolution of unburied pipe under stochstic P nd for nonliner dge.
For buried pipes (figure 4.43) the lifetie is nerly 7.49 yers for ode (High pressure),.9 yers for ode (Middle pressure), nd.64 yers for ode 3 (Low pressure). The degrdtion curves show lso ore steep evolution for the two first odes thn the third ode. Figure 4.43 - Degrdtion evolution of buried pipe under stochstic P nd for nonliner dge. The lifeties for offshore pipes (figure 4.44) show tht is nerly 9.5 yers for ode (High pressure), 6.4 yers for ode (Middle pressure), nd 8.7 yers for ode 3 (Low pressure). The degrdtion evolutions re steeper for the two first odes thn the third ode. Figure 4.44 - Degrdtion evolution of offshore pipe under stochstic P nd for nonliner dge. 97
The stochstic influence cn be seen through the vribility over the curve reliztions of D(t) obtined by ny siultions nd not fro just one reliztion. ontrrily to the cse of one rndo vrible, the curves re not sooth nd the stochstic effects re clerer here. To ore exploit these results, en curve D(t) cn be plotted fro the en vlue of these reliztions. The conservtive curves re those tht give the xiu vlues. For ech ode, chrcteristic curve of lifetie cn be coputed fro the en vlues, the stndrd devition vlues, nd certin frctl percentge depending on the risk dopted by decision kers. IV..4.. - oprison: Deterinistic - Stochstic Results (onliner Dge Lw) To show the stochstic effects, coprison is done between the deterinistic results nd the stochstic results (figure 4.45). Figure 4.45 - Deterinistic nd stochstic (P, ) study of offshore pipes degrdtion under nonliner dge lw. Deterinistic nonliner Stochstic nonliner Decrese (%) Mode.9 yers 9.5 yers 5.3% Mode 9. yers 6.4 yers 4.% Mode 3 33.67 yers 8.7 yers 4.7% 98
For ll odes of internl pressure, the lifeties of pipes decrese bout 5% fro the deterinistic cse to the stochstic cse. These reductions re explined by the fct tht the dispersions introduced by the rndo vribles hve negtive effect on the lifeties' predictions. The stochstic effect is ore pronounced nd effective for two rndo vribles thn for one rndo vrible. The curves for ech ode fluctute nd they constitute bundle of trjectories which re the reliztions of ny siultions. IV..5 - Vlidtion of the Pipelines Lifeties in Stochstic onditions The obtined lifeties vlues for liner nd nonliner dge rules in stochstic conditions cn be verified to be in the rnge of rel lifeties ccording to the references [34,35,36]. In fct, ftigue life of pipes under good exploittion conditions ws found to be 6 yers in verge which is very close to the results obtined for pipes in ode 3 in stochstic nonliner cse. 99
IV. - onclusion In this chpter the prognostic odel is developed to consider the prognostic coputtion in stochstic conditions. Hence, the odel is generl one s it is bsed on the liner nd nonliner ccuultion of dge due to ftigue crck propgtion in stochstic conditions. These lst conditions re tken into ccount by considering two rndo vribles which re the pplied loding nd the initil crck length. Two cses re explored seprtely: one rndo vrible nd two rndo vribles. The ftigue filure is considered nd the dge stte of the device is esured by degrdtion indictor in ters of the nuber of loding cycles strting fro n initil dge. The lifeties re concluded fro the tie reding t ech instnt on the degrdtion curve. The Reining Useful Lifeties t ech instnt re deduced fro the degrdtion curve by subtrcting the current instnt fro the lst predicted instnt. To show the efficiency of this stochstic prognostic odel, it is pplied to predict the ftigue life of vehicle suspension systes nd of petrocheicl pipelines under three odes of internl pressure. Lifeties results re obtined for liner nd nonliner stochstic cses. The stochstic results for one rndo vrible show tht the nonliner cse is lwys doinnt where the curves re not fluctunt. ontrrily, for two rndo vribles cse the stochstic effects becoe ore influent nd the curves of degrdtion re fluctunt nd constituted of bundles of trjectories. In this cse the lifeties re reduced due to the dispersion effects.
References [] A.K.S. JARDIE. D.L. nd D. BAJEVI: A review on chinery dignostics nd prognostics ipleenting condition-bsed intennce. Mechnicl Systes nd Signl Processing, (7): 483-5, 6. [] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, nd M. OULADSIE, "Anlytic nd onliner Prognostic for Vehicle Suspension Systes," IEEE Interntionl onference on Prognostic nd Helth Mngeent (PHM ), Denver, olordo, USA, June 8-,. [3] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, nd M. OULADSIE, "Stochstic Prognostic Prdig for Petrocheicl Pipelines Subject to Ftigue," Interntionl Journl of Systes Science (IJSS), Tylor & Frncis publishers, London,, subitted. [4] A. HESS, G. ALVELLO, P. FRITH, S. EGEL, nd D. HOITSMA, "hllenges, Issues, nd Lessons Lerned hsing the 'Big P': Rel Predictive Prognostics Prt," Proc. IEEE 489 (), 5. [5] S. MOHATY, S. DAS, A. HATTOPADHYAY, P. PERALTA, nd. WILLHAUK, "Tie Series Prediction of Ftigue rck Growth Using Multi-Vrite Gussin Processes," Interntionl Journl of Ftigue, 8. [6] J.M. LARSE nd L. HRISTODOULOU, "Integrted Dge Stte Awreness nd Mechnis-Bsed Prediction," Journl of Metls, TMS, p.4, Mrch 4. [7] L. HRISTODOULOU nd J.M. LARSE, "Using Mterils Prognosis to Mxiize the Utiliztion Potentil of oplex Mechnicl Systes," Journl of Metls, TMS, pp. 5-9, Mrch 4. [8] S.J. HUDAK, JR., M.P. ERIGHT, R.. MLUG, H.R. MILLWATER, A. SARLASHKAR, nd M.J. ROEMER, "Potentil Benefits of Adding Probbilistic Dge Accuultion to Prognosis of Turbine Engine Relibility," SwRI Finl Report to AFRL/DARPA, ontrct o. F3365-97-D-57, June 3,. [9] G.R. LEVERAT, D.L. LITTLEFIELD, R.. MLUG, H.R. MILLWATER, nd J.Y. WU, "A Probbilistic Approch to Aircrft Turbine Mteril Design," Pper 97-GT-, ASME Interntionl Gs Turbine & Aeroengine ongress, 977. [] M.P. ERIGHT, Y.D. LEE, R.. MLUG, L. HUYSE, G.R. LEVERAT, H.R. MILLWATER, nd S.K. FITH, "Probbilistic Surfce Dge Tolernce Assessent of Aircrft Turbine Rotors," Pper GT-3-3873, Proceedings, 48 th ASME Interntionl Gs Turbine & Aeroengine Technicl ongress, Atlnt, GA, June 3. [] J.. YAG nd S.D. MAIG, "Stochstic rck Growth Anlysis Methodologies for Metllic Structures," Engineering Frcture Mechnics, 37 (5),5-4, 99. [] J.. YAG nd S.D. MAIG, "A Siple Second Order Approxition for Stochstic rck Growth Anlysis," Probbilistic Engineering Mechnics, 8, 7-8, 3.
[3] W.F. WU nd.. I, "A Study of Stochstic Ftigue rck Growth Modeling through Experientl Dt," Probbilistic Engineering Mechnics, 8,7-8, 3. [4] W.F. WU nd.. I, "Probbilistic Models of Ftigue rck Propgtion nd Their Experientl Verifiction," Probbilistic Engineering Mechnics 9,47-57, 4. [5] K. SOBZYK, nd B.F. SPEER, "Rndo Ftigue: fro Dt to Theory," Acdeic press, Boston, MA, 99. [6] J.L. BOGDAOFF nd F. KOZI, "Probbilistic Models of uultive Dge," Wiley, ew York, 985. [7] F. KOZI nd J.L. BOGDAOFF, "Probbilistic Models of Ftigue rck Growth: Results nd Specultions," ucler Engineering nd Design 5, 43-7, 989. [8] W.Q. ZHU, Y.K. LI, nd Y. LEI, "On Ftigue rck Growth under Rndo Loding," Engineering Frcture Mechnics 43 (), -, 99. [9]. WILLHAUK, S. MOHATY, A. HATTOPADHYAY, nd P. PERALTA, "Stochstic rck Growth under Vrible Loding for Helth Monitoring nd Prognosis," Proceeding of SPIE 696 696L, Sn Diego, liforni, USA, 8. [] J.. YAG, S.D. MAIG, J.L. RUDD, nd M.E. ARTLEY, "Probbilistic Durbility Anlysis Methods for Metllic Airfres," Probbilistic Engineering Mechnics (), 9-5, 987. [] Y.J., HOG, J. XIG, nd J.B. WAG, "Sttisticl Anlysis of Ftigue rck Growth for 6MnR Steel under onstnt Aplitude Lods," Interntionl Journl of Pressure Vessels nd Piping 76, 379-385, 999. [] B. MOREO, J. ZAPATERO, nd J., DOMIGUEZ, "An Experientl Anlysis of Ftigue rck Growth under Rndo Loding," Interntionl Journl of Ftigue 5, 597-68, 3. [3] T.K. SERGEEVA, A.. BOLOTOV, et l., "Monitoring of Steel ondition in Min Pipelines during Their Stress-orrosion Induced Filures," heicl nd oil Mchinery, o., pp. 7-76, 996. [4].E. JASKE, "Fitness-for-service Assessent for Pipelines Subject to Stresscorrosion crcking," The pipeline pigging nd integrity conference, Februry. [5] S.A. TIMASHEV, M.G. MALYUKOVA, nd, LL. MALTSEV, "Updting the Assessent of Reining Life of Pipelines using Ltest ILI Dt nd the Iportnce Spling Method," Science nd Engineering center, Relibility nd resource of lrge chine systes, Url Brnch, Russin cdey of sciences, Rotterd, IOSSAR 5. [6] P. PARIS, F. ERDOGA, "A riticl Anlysis of rck Propgtion Lws," Journl of Bsic Engineering, Trnsctions of the Aericn Society of Mechnicl Engineers, Vol. 85, o. 4, pp. 58-534, 963.
[7] A. ABOU JAOUDE, S. KADRY, K. EL-TAWIL, H. OURA, nd M. OULADSIE, "Anlytic Prognostic for Petrocheicl Pipelines," Journl of Mechnicl Engineering Reserch (JMER), Vol. 3(3), pp. 64-74, Mrch. [8] A. ABOU JAOUDE, K. EL-TAWIL, S. KADRY, H. OURA, nd M. OULADSIE, "Prognostic Model for Buried Tubes," Interntionl onference on Advnced Reserch nd Applictions in Mechnicl Engineering (IARAME'), otre De University, Louize, Lebnon, June 3-5,. [9] M.A. MIER, uultive Dge in Ftigue, Journl of Applied Mechnics, vol., A59-A64, 945. [3] J. LEMAITRE nd J. HABOHE, Mechnics of Solid Mterils. ew York: bridge University Press, 99. [3] G. PETRUI nd B ZUARELLO, "Ftigue life prediction under wide bnd rndo loding," Diprtiento di Meccnic, Universit degli Studi di Plero, Vile delle Scienze, Itly, April 4. [3] J. MADSE, D. GHIOEL, D. GORSIH, D. LAMB, nd D. EGRUT,"A Stochstic Approch to Integrted Relibility Prediction," University of Wisconsin-Mdison, June 4, 9 [33] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, nd M. OULADSIE, "Anlytic Prognostic Model for Stochstic Ftigue of Petrocheicl Pipelines," Austrlin ontrol onference (AU ), Sydney, Austrli, oveber 5-6,. [34] X. GUA, R. JHA, nd Y. LIU, "Trns-diensionl MM for Ftigue Prognosis Model Deterintion, Updting, nd Averging," Annul onference of the Prognostic nd Helth Mngeent society,. [35] Y. XIAG nd Y. LIU, "Efficient Probbilistic Methods for Rel-tie Ftigue Dge Prognosis," Annul onference of the Prognostic nd Helth Mngeent Society,. [36] R. PALMER-JOES nd T. E. TURER, "Pipeline Buckling, orrosion nd Low ycle Ftigue," Offshore Mechnics nd Arctic Engineering, OMAE 98 Lisbon, United Kingdo July 5-9, 998. 3
OLUSIO nd FUTURE WORKS A prognostic odel is introduced in this thesis tht perits to predict the degrdtion trjectory of dynic syste; it is bsed firstly on nlyticl lws of dge such s the crck propgtion lw nd liner dge ccuultion lw. Secondly, it is bsed on nonliner dge ccuultion nd finlly, the stochstic influences re considered. In the pproches bsed on physicl or theticl odels, the knowledge of the fundentl equtions of the dynic behvior of degrdtion ppers to be very useful. In fct, in cse we chnge the syste properties or of degrdtion, the preters cn be redjusted nd then the pproch is dptble to new cse. The pproches guided by dt ssue relible estition of the current stte of degrdtion in order to predict the future evolution of the syste. They lck rectivity when fcing chnge in usge conditions nd the efficiency is strongly linked to the sple of dt tht serves to copute the odel preters. The third pproch which is the Experience-bsed pproch requires little expert knowledge of the degrdtion echniss. It reins siple to ipleent but it is lso insensitive to chnge in the syste operting ode. In ddition, the odels derived hve only two sttes: stte of functioning, nd stte of filure, nd do not coprise stte of degrded functioning. The proposed odel belongs to the first prognostic pproch which is the odelbsed pproch. Whenever the nlytic dge lws re vilble, this odel cn be dptble to new situtions or cses. In industril systes, this odel shows tht it is convenient nd prcticl s flexible tool for prognostic nlysis. The filure ode treted in this thesis is the ftigue of the device teril. The considered dge is the crck propgtion due to ftigue. The dge stte of the device is esured by degrdtion indictor D in ters of the nuber of loding cycles. The proposed odel is bsed on the link between conventionl index of degrdtion D tht vries fro zero to one nd the crck length. A filure is produced when reches criticl length. The odel is then expressed by recursive function relting the degrdtion in two consecutive cycles to the criticl nuber of cycles nd the endurnce stress liit of the 5
teril. Fro detected initil crck, the degrdtion trjectories hve been drwn in ters of cycle loding. The nlytic prognostic odel introduced in this thesis perits to predict, t ech cycle or instnt, the reining useful lifetie of the syste by siple nd prcticl wy. The lifeties re concluded fro the tie reding t ech instnt on the degrdtion curves or trjectories. To show the efficiency of this prognostic odel, it is pplied in siultion to predict the ftigue life of the petrocheicl pipeline systes nd of the vehicle suspension systes. In fct, the degrdtion trjectories deduced llow us to deterine their reining useful lifeties. There re ny cuses nd contributors to pipelines filures, including construction errors, teril defects, pressure fluctutions, gs blows, internl nd externl corrosion, opertionl errors, lfunction of control systes nd outside force dge (e.g., by third prties during excvtion). Pipeline incidents cn result in loss of life, serious injury, property dge, nd environentl dge, lthough jor incidents re infrequent. In ny cses, pipelines plced underground, under runwys or rodwys re required to resist the influence of the overlying soil nd ny surfce trffic lods ccidents s well s the effect of corrosion nd teril filure like ftigue. For these resons, the ftigue life prediction is done for unburied, buried nd offshore pipelines under three odes of internl pressure. Additionlly, nonliner interction effect exists between high cycle ftigue (HF) nd low cycle ftigue (LF) in ny engineering terils. It hs been observed within unixil lodings, nd ore pronounced under ultixil loding, prticulrly when the loding is non-proportionl. This nonliner odeling is especilly iportnt to tke into ccount the nture of the pplied constrints nd influent environent tht cn ccentute the nonliner spect relted to soe terils behvior subject to ftigue effects. In the proposed nonliner ccuultion of dge, the dge stte of the device is esured by recursive nonliner degrdtion function in ters of the nuber of cycles or usge tie. This nonliner prognostic odel is pplied to estite the ftigue life of pipeline syste nd vehicle suspension syste in order to revel the effectiveness of this odel. The RUL results obtined re copred to previous results of liner odel nd the 6
differences re justified by the ultiple trends of degrdtion (liner, convex, nd concve). The present nonliner prognostic odel will llow us to include the stochstic spect which will iprove the intended prediction cpcity of the odel. In the extended stochstic odel, bsed on the ccuultion of dge due to ftigue crck propgtion in stochstic conditions, the initil crck length nd the loding re tken s rndo. The prognostic odel becoes ore precise in RUL prediction. Lifetie results re obtined for liner nd nonliner dge cses nd the differences re justified by the ultiple trends of degrdtion lso. Stochstic crck propgtion involves odels with rndo preters which cn be estited using Monte rlo siultions. The stochstic preters re ffected by soe probbility of reliztion tht influences the resulting RUL deduced fro the degrdtion trjectory. As prospective nd future works, it is plnned to ore develop the proposed prognostic ethodology nd pply it to wide set of dynic systes. This is by tking into considertion other nlytic lws besides Pris-Erdogn's lw for crck propgtion nd other dge ccuultion lws. Additionlly, ore probbilistic bsic preters like the teril nd the environentl preters cn be considered. Furtherore, dditionl probbilistic lws for the preters other thn the orl nd the Log-norl lws cn be explored. Also, it is plnned to ore explore the vribility of the stochstic lifeties nd to deduce bundle of degrdtion curves fro which en curve nd chrcteristic lifetie curve cn be inferred. The chrcteristic curve is the one ttched to soe predefined cceptble risk. As well, in the pipeline ppliction, other internl pressure odel fluctution cn be tken into ccount s for exple the odel of the Fourier series. In the utootive suspension syste, the output vribles (verticl displceents of dpers) cn be derived fro the input vribles (rod profile). This is done by resolution of convenient dynic odel by considering the inertil forces which re due to the vehicle oscilltory oveent on rod with n irregulr surfce. 7
LIST OF PUBLIATIOS Interntionl Journls [] A. ABOU JAOUDE, K. EL-TAWIL, S. KADRY, "Prediction in oplex Diension Using Kologorov s Set of Axios", Journl of Mthetics nd Sttistics, vol. 6(), pp. 6-4,. [] A. ABOU JAOUDE, K. EL-TAWIL, S. KADRY, H. OURA, nd M. OULADSIE, Anlytic Prognostic Model for Dynic Syste, Interntionl Review of Autotic ontrol (IREAO), oveber. [3] A. ABOU JAOUDE, S. KADRY, K. EL-TAWIL, H. OURA, nd M. OULADSIE, "Anlytic Prognostic for Petrocheicl Pipelines", Journl of Mechnicl Engineering Reserch (JMER), April. [4] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, nd M. OULADSIE, "Stochstic Prognostic Prdig for Petrocheicl Pipelines Subject to Ftigue", Interntionl Journl of Systes Science (IJSS), Tylor & Frncis publishers, London,, subitted. Interntionl onferences [] K. EL-TAWIL, A. ABOU JAOUDE, S. KADRY, H. OURA, nd M. OULADSIE, "Prognostic Bsed on Anlytic Lws Applied to Petrocheicl Pipelines", Interntionl onference on oputer-ided Mnufcturing nd Design (MD ), hin, oveber. [] A. ABOU JAOUDE, K. EL-TAWIL, S. KADRY, H. OURA, nd M. OULADSIE, Prognostic Model for Buried Tubes, Interntionl onference on Advnced Reserch nd Applictions in Mechnicl Engineering (IARAME'), otre De University, Louize, Lebnon, June 3-5,. [3] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, AD M. OULADSIE, "Lifetie Anlytic Prognostic for Petrocheicl Pipes Subject to Ftigue", SAFEPROESS, 8th IFA Syposiu on Fult Detection, Supervision nd Sfety of Technicl Processes, Mexico ity, Mexico, August 9-3,. [4] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, nd M. OULADSIE, "Anlytic nd onliner Prognostic for Vehicle Suspension Systes", IEEE Interntionl onference on Prognostic nd Helth Mngeent (PHM ), Denver, olordo, USA, June 8-,. [5] A. ABOU JAOUDE, H. OURA, K. EL-TAWIL, S. KADRY, nd M. OULADSIE, "Anlytic Prognostic Model for Stochstic Ftigue of Petrocheicl Pipelines", Austrlin ontrol onference (AU ), Sydney, Austrli, oveber 5-6,. 9
THESIS ABSTRATS Advnced Anlyticl Model for the Prognostic of Industril Systes Subject to Ftigue The high vilbility of technologicl systes like erospce, defense, petrocheistry nd utoobile, is n iportnt gol of erlier recent developents in syste design technology knowing tht the expensive filure cn generlly occur suddenly. To ke the clssicl strtegies of intennce ore efficient nd to tke into ccount the evolving product stte nd environent, new nlytic prognostic odel is developed s copleent of existent intennce strtegies. This new odel is pplied to echnicl systes tht re subject to ftigue filure under repetitive cyclic loding. Knowing tht, the ftigue effects will initite icro-crcks tht cn propgte suddenly nd led to filure. This odel is bsed on existing dge lws in frcture echnics, such s the crck propgtion lw of Pris-Erdogn beside the dge ccuultion lw of Plgren-Miner. Fro predefined threshold of degrdtion D, the Reining Useful Lifetie (RUL) is estited by this prognostic odel. Dges cn be ssued to be ccuulted linerly (Plgren-Miner's lw) nd lso nonlinerly to tke into considertion the ore coplex behvior of loding nd terils. The degrdtion odel developed in this work is bsed on the ccuultion of dge esureent D fter ech loding cycle. When this esure reches the predefined threshold D, the syste is considered in wer out stte. Furtherore, the stochstic influence is included to ke the odel ore ccurte nd relistic. In this work, two in pplictions re considered: in utoobile industry, prognostic ssessent of the suspension coponent perits to enhnce its intennce strtegies; nd in petrocheicl industries, pipelines re studied to prevent the sudden nd hrful lekge or blows. Keywords: Prognostic, Reining Useful Lifetie, Ftigue, Degrdtion, Anlytic odel, Liner ccuultion, onliner ccuultion, Dge, Stochstic.
Modèle Anlytique Avncé pour le Pronostic des Systèes Industriels Souis à l Ftigue L disponibilité élevée des systèes technologiques coe l'érosptil, l défense, l pétrochiie et l'utoobile, est un but iportnt des nouveux développeents de l technologie de conception des systèes schnt que l défillnce onéreuse survient, en générl, soudineent. Afin de rendre les strtégies clssiques de intennce plus efficces et pour prendre en considértion l'étt et l'environneent évolutifs du produit, un nouveu odèle de pronostic nlytique est développé en tnt que copléent des strtégies de intennce existntes. e nouveu odèle est ppliqué ux systèes écniques souis à l défillnce pr ftigue sous chrge cyclique répétitive. Schnt que l'effet de ftigue v initier des icrofissures qui peuvent se propger soudineent et conduire à l défillnce. e odèle est bsé sur des lois d'endogeent existntes dns l écnique de l rupture coe l loi de propgtion de fissures de Pris-Erdogn à côté de l loi de cuul de doge de Plgren-Miner. A prtir d'un seuil prédéfini de dégrdtion D, l durée de vie résiduelle (RUL) est estiée à l'ide de ce odèle de pronostic. Les doges peuvent être cuulés linéireent (Loi de Plgren-Miner) et ussi non linéireent fin de prendre en copte un coporteent plus coplexe des chrgeents et des tériux. Le odèle de dégrdtion développé dns ce trvil est bsé sur une sotion d'une esure de doge D à l suite de chque cycle de chrgeent. Qund cette esure devient égle à un seuil prédéfini D, le systèe est considéré dns l'étt de pnne. En plus, l'influence stochstique est incluse dns notre odèle pour le rendre plus précis et réliste. Dns ce trvil, deux pplictions principles sont considérées: dns l'industrie utoobile, l'évlution de pronostic des éléents de suspension peret d'éliorer ses strtégies de intennce; et dns l'industrie pétrochiique, les pipelines sont étudiés fin de prévenir des fuites et des explosions soudines et nocives. Mots-clefs: Pronostic, Durée de vie résiduelle, Ftigue, Dégrdtion, Modèle nlytique, uul linéire, uul non-linéire, Doge, Stochstique.
RÉSUMÉ DE LA THÈSE Modèle Anlytique Avncé pour le Pronostic des Systèes Industriels Souis à l Ftigue L disponibilité élevée des systèes technologiques coe l'érosptil, l défense, l pétrochiie et l'utoobile, est un but crucil des nouveux développeents de l technologie de conception des systèes. En générl, l défillnce est onéreuse et elle survient soudineent. Le pronostic consiste en l cpcité de ''prévoir et prévenir'' des défuts possibles ou de l dégrdtion du systèe vnt l'occurrence des pnnes. S'il étit possible de prédire efficceent l'étt des chines et des systèes, les ctions de intennce peuvent être exécutées u bon oent. Le pronostic est défini coe "prédire l défillnce qund elle survient", utreent, prvenir à un oyen de clcul de l durée de vie résiduelle d'un coposnt. Afin d'obtenir un pronostic efficce et fible, il est nécessire d'voir un dignostic efficce et fible. Au sens Autotique du tere, le pronostic est générleent ssocié à l notion de dégrdtion qui représente le cuul de l'usure d'un systèe. Il consiste à prévoir l future évolution de l dégrdtion en prennt en considértion les fcteurs qui odifient les dyniques de l dégrdtion. es fcteurs peuvent être divisés en deux ctégories: les fcteurs liés à l sollicittion du systèe et ceux liés à l'environneent dns lequel le systèe évolue. orleent, l'influence de ces deux ctégories sur l dégrdtion n'est ps bien connue. oe les strtégies clssiques de intennce peuvent être éliorées puisqu'elles négligent l'étt et l'environneent évolutifs du produit, lors les pproches de pronostic ont prouvé leurs intérêts dns ce doine. Différentes éthodes ont été ppliquées u pronostic des coposnts dégrdés. En générl, les pproches de pronostic peuvent être clssifiées en trois ctégories fondentles: 3
() Approches "à bse de odèles", () Approches "guidées pr les données", et (3) Approches bsées sur les techniques probbilistes. L'vntge principl des pproches "à bse de odèles" est leur cpcité à inclure les infortions physiques du systèe surveillé. De êe, si les infortions recueillies de l dégrdtion du systèe deviennent plus disponibles, lors le odèle de pronostic peut être rédpté pour prendre en copte ces nouvelles infortions fin d'ugenter s précision de prédiction et de triter des problèes de perfornce plus délicts. ependnt, les pproches "guidées pr les données" s'ppliquent lorsque le odèle n'existe ps is elles nécessitent un nobre suffisnt de esures de bonnes qulités fin de bien refléter l'ige de dégrdtion du systèe. Les pproches bsées sur les techniques probbilistes nécessitent un excellent retour d'expérience (historique, données expertes, etc.) perettnt une odélistion stochstique ou probbiliste de l dégrdtion. es pproches sont bien dptées ux systèes coplexes pour lesquels il est difficile d'voir un odèle physique. Une nouvelle procédure nlytique de pronostic "à bse de odèles" est développée dns cette thèse et ppliquée ux systèes écniques souis à l ftigue sous chrge cyclique répétitive; schnt que les effets de l ftigue initieront des icrofissures qui peuvent se propger soudineent et conduire à l défillnce. e odèle est bsé sur des lois d'endogeent existntes dns l écnique de l rupture coe l loi de propgtion de fissures de Pris-Erdogn à côté de l loi de cuul de doge de Plgren-Miner. A prtir d'un seuil prédéfini de dégrdtion D, l durée de vie résiduelle (RUL) est estiée à l'ide de ce odèle de pronostic. Les doges peuvent être cuulés linéireent (Loi de Miner) et ussi non linéireent fin de prendre en copte un coporteent plus coplexe. ette thèse est dédiée u pronostic des systèes dyniques. Les trvux de cette thèse ont pour but le développeent d'un outil vncé perettnt de triter l'évlution du pronostic dns un contexte déteriniste linéire et non-linéire dns un preier teps, et 4
dns un contexte stochstique dns un second teps. otre objectif est de préprer un oyen générl de pronostic cpble de bien prédire l durée de vie résiduelle (RUL) d'un systèe. ette prédiction est bsée sur un cuul nlytique de doge et ceci dns les deux contextes déteriniste et stochstique. otre odèle de dégrdtion est fondé sur un cuul d'une esure de doge D à l suite de chque cycle de chrgeent. Qund cette soe devient égle à D, le systèe est considéré dns un étt de pnne. En plus, l'effet stochstique est inclus dns notre odèle pour le rendre plus précis. Dns ce trvil, deux pplictions principles sont considérées: dns l'industrie utoobile où l'évlution de pronostic des éléents de suspension peret d'éliorer ses strtégies de intennce; et dns l'industrie pétrochiique dns lquelle les pipelines sont étudiés fin de prévoir des éventuelles fuites et des explosions soudines et nocives. Le preier chpitre est conscré à l littérture et à l'étt de l'rt générl de l science de pronostic. Il décrit pleent les différentes pproches proposées dns ce doine pr les spécilistes de pronostic. En effet, dns ce preier chpitre, un tour d'horizon coplet des pproches de pronostic est présenté, ussi bien que les vntges et les inconvénients de chcune des trois filles de pronostic sont bordés. Il ontre l grnde iportnce de ces genres d'étude pour les systèes technologiques et industriels. L éthodologie bsée sur les bques de dégrdtion est discutée. Elle ontré l'iportnce de cette nouvelle pproche qui peret de suronter les inconvénients des odèles de pronostic existnts à conditions d'voir un grnd nobre de données disponibles et fibles. Le problèe principl de l'pproche bsée sur l'expérience est qu'elle ne peut ps être ppliquée dns le cs des nouveux systèes pour lesquels les données collectées pr retour d'expérience n'existent ps ou s'vèrent insuffisntes. Les pproches guidées pr les données s'ppuient sur une estition fible de l'ige de l'étt cournt de dégrdtion fin de prédire l future évolution du systèe. L'efficcité des éthodes d'pprentissge est liée forteent à l'échntillon des données qui sert à clculer les 5
prètres du odèle. Si une sitution non pprise surviendr, le pronostic peut être létoire. De êe, les pproches guidées pr les données nquent de réctivité fce à des chngeents dns les conditions d'utilistion. Qund les pproches sont dépourvues des fores nlytiques, elles ontrent souvent une inflexibilité durnt l'ppliction à des coporteents vriés des systèes. L'pproche de pronostic bsée sur l'expérience nécessite peu de connissnce experte des écnises de dégrdtion. Elle reste fcile à ettre en œuvre is elle n'est ps réctive fce à l'éventuel chngeent dns le ode de fonctionneent du systèe. En plus, les odèles construits dns cette pproche, ont seuleent deux étts: un étt de fonctionneent et un étt de défillnce, ils ne coprennent ps un étt de fonctionneent dégrdé. Dns les pproches bsées sur les odèles thétiques ou physiques, l connissnce des équtions du coporteent dynique de l dégrdtion s'vère très utile. En cs de chngeent des propriétés du systèe ou de l dégrdtion, les prètres peuvent être réjustés et le odèle peut être rédpté à un nouveu cs. ependnt, il est nécessire d'voir une hute qulifiction fin de bien itriser les écnises de dégrdtion en question, d'où le coût élevé de l'utilistion ce type de odèle. énoins, l précision et l'exctitude recherchées éritent le surcoût pyé. Donc le choix d'une nouvelle pproche à bse physique, fondée sur un nouveu odèle thétique de dégrdtion, devient logique et justifié. Pr suite, des lois thétiques précises, utiles et élégntes nous ideront dns les chpitres qui suivent fin d'chever le but de cette thèse. otre odèle propose l'utilistion des lois nlytiques de doge. Le deuxièe chpitre définit le critère dopté, à svoir l rupture pr ftigue, et développe un odèle bsé sur l'spect linéire de cuul de doge. Le odèle de pronostic proposé dns cette thèse peret de prédire l trjectoire de dégrdtion d'un systèe dynique; il est bsé, preièreent, sur des lois nlytiques de doge à cuul linéire déjà évoquées, deuxièeent, il est bsé sur une loi de cuul non-linéire de doge (troisièe chpitre) et troisièeent, il fit inclure les influences stochstiques (qutrièe chpitre). 6
L loi de Pris nous peris de odéliser l'évolution de l longueur de fissure vec le nobre de cycles de chrgeent dns l phse stble de propgtion. A chque cycle, l longueur de fissure subit un incréent; et qund cette longueur tteint une certine vleur critique, u-delà de lquelle l rupture devient iinente, l pièce est déclrée en étt défectueux. L esure de dégrdtion doptée est un sclire D norlisé vrint entre et et relié u nobre de cycles à trvers l loi de Miner en profitnt de l propriété d'dditivité linéire de cette loi. Le ode de défillnce trité dns ce trvil est l ftigue des tériux du dispositif. Le doge considéré est dû à l propgtion de fissure pr ftigue. L'étt d'endogeent du dispositif est esuré pr un indice de dégrdtion D en fonction du nobre de cycles de chrgeent. Le odèle proposé est bsé sur une reltion entre un indice conventionnel de dégrdtion D et une longueur de fissure. L défillnce ser déclrée qund tteint l longueur critique. Le odèle est donc exprié pr une fonction linéire récursive relint l dégrdtion dns deux cycles consécutifs u nobre critique de cycles et à l contrinte liite d'endurnce du tériu du systèe. A prtir d'une fissure initile détectée, les trjectoires de dégrdtion peuvent être trcées en fonction de cycles de chrgeent. Le odèle nlytique de pronostic développé dns cette thèse peret de prédire, à chque cycle ou instnt, l durée de vie résiduelle (RUL) du systèe. Les durées de vie sont déduites à prtir d'une lecture de teps, en chque point, sur les courbes et les trjectoires de dégrdtion obtenues. e odèle pprtient à l preière fille des pproches de pronostic. Dns le cs où les lois nlytiques de doge sont disponibles, ce odèle est qulifié d'dptble ux nouvelles situtions. A notre vis, ce odèle perettr d'ssure un oyen utile pour l'nlyse de pronostic des systèes industriels. Afin d'illustrer l éthodologie présentée et de ontrer son efficcité, l'pproche proposée est ppliquée à l prédiction de l'âge des deux systèes en ftigue. L'étude considère preièreent l'ppliction industrielle à un systèe de suspension d'utoobile, et deuxièeent, l'ppliction à un systèe pétrochiique coe les pipelines. Dns ces deux 7
pplictions, des courbes de dégrdtions sont déduites perettnt insi de déteriner les durées de vie des éléents industriels étudiés. On considère dns notre ppliction un systèe foré de l oitié d'une suspension à cuse de l syétrie. Les suspensions sont souises à un chrgeent répété dû à l surfce d'une route non régulière. ette surfce est odélisée pr une fonction polynoile périodique. Trois odes d'excittion de route sont exinés en fonction de l'plitude de l surfce odélisée fin de tenir copte des cs extrêes d'étt de route et du fonctionneent de l suspension. En ce qui concerne l deuxièe ppliction, l'iportnce de l'étude du pronostic des pipelines réside dns le fit qu'il existe plusieurs origines de l défillnce de ces tuyux, y copris: erreurs de construction, défuts de tériux, fluctution de pression, explosion de gz, corrosion interne et externe, erreurs opértionnelles, dysfonctionneent des systèes de contrôle et force d'endogeent extérieure (issue d'un tiers durnt l'excvtion). Les ccidents des pipelines peuvent conduire à des pertes de vie, à des blessures grves, à l'endogeent des propriétés et à l nuisnce à l'environneent bien que les ccidents jeurs sont rres. Dns plusieurs cs, les tuyux plcés sous terre, sous routes et sous utoroutes sont supposés résistnts à l'influence des couches supérieures du sol et de plusieurs chrgeents routiers de trfic, ussi bien à l'effet de l corrosion et de l rupture de tériu pr ftigue. Pour toutes ces risons, l prédiction de vie en ftigue est effectuée pour des tuyux vec leurs trois odes de plceent: à surfce, enterrés, et offshore (sous-rins). En plus, trois odes de pressions internes sont pris en copte fin d'explorer les cs extrêes de fonctionneent. Dns le chpitre trois, nous introduisons une loi non linéire pour le cuul de doge à l plce de l loi linéire de Miner. L'iportnce de cette éliortion réside dns le fit qu'un effet non linéire d'interction existe entre l ftigue à hut cycle (HF) et l ftigue à bs cycle (LF) dns plusieurs tériux utilisés surtout en génie écnique. ette non-linérité est observée dns le chrgeent uni-xil et, encore plus prononcée, dns le chrgeent ultixil. eci existe prticulièreent qund le chrgeent est non 8
proportionnel. En plus, cette odélistion non-linéire est encore iportnte puisqu'elle prend en copte l nture des contrintes ppliquées et l'environneent influnt. e dernier peut ccentuer encore plus l'spect non-linéire reltif ux certins coporteents de tériux sous l'effet de l ftigue. En plus, des éthodes trditionnelles de cuul de doge ont ontré une prédiction de vie iprécise qund des niveux de chrge ultiples sont siultnéent considérés. Dns le odèle proposé ici, bsé sur un cuul non-linéire de doge, l'étt d'endogeent du dispositif est esuré à chque cycle pr une fonction récursive nonlinéire de dégrdtion en fonction des rges des contrintes ppliquées et du nobre de cycles de chrgeent ou du teps écoulé de fonctionneent. ette fonction récursive est déduite d'une résolution d'une éqution différentielle ordinire du preier ordre inclunt l dérivée de l dégrdtion pr rpport u nobre de cycles en fonction de contrintes de chrgeent, des prètres des tériux et de l'environneent, du nobre critique de cycles, de l'endurnce et de l dégrdtion instntnée. Afin de ontrer l'efficcité de ce odèle non-linéire, il est ppliqué pour prédire l vie en ftigue du systèe de suspension d'utoobile et du systèe des tuyux. Les résultts du clcul de l durée de vie résiduelle (RUL) sont coprés ux résultts issus du odèle linéire et l'écrt est justifié pr les différentes tendnces de dégrdtion (linéire, convexe et concve). Dns les pplictions effectuées, les résultts optiistes du cs non-linéire peuvent être expliqués pr le fit que qund les tendnces réelles de dégrdtion (non-linéires) sont de fores concves, lors le cuul de doge est surestié qund une fore linéire est utilisée à l plce d'une fore non-linéire. Dns l'ppliction ux pipelines, l'étude non-linéire seble fournir un coporteent de doge plus réliste pour les différentes vleurs de pression reltiveent u cs linéire. En effet, contrireent u cs linéire, le cs non-linéire présente une nette différence entre les trois odes de pression qund on s'pproche de l'étt de défillnce. e odèle de 9
pronostic non-linéire fcilite l'introduction de l'spect stochstique qui éliorer l cpcité prédictive du odèle proposé. Le qutrièe chpitre étend le prdige déteriniste développé dns cette thèse u doine stochstique. Les outils de pronostic de défillnce doivent voir l cpcité d'inclure le doge des tériux sous des conditions de fonctionneent norles et extrêes. Le odèle s'ppuie sur un cuul de doge dû à l propgtion des fissures de ftigue dns des conditions probbilistes. L longueur initile de fissure et le chrgeent ppliqué sont considérés lors létoires. En plus, l durée de vie résiduelle (RUL) peut être expriée en ftigue sous plusieurs fores: soit l longueur critique de l fissure soit le nobre critique de cycles de chrgeent soit l téncité des tériux K I. ous pouvons écrire lors différents étts liites ou différents critères de perfornce qui ne sont que les rges entre une esure instntnée de doge intrinsèque et une vleur liite (critique) à ne ps dépsser. Plusieurs étts liites peuvent être lors considérés et rendus létoires si leurs vribles de bse sont probbilistes. Des incertitudes considérbles existent dns l'utilistion et dns les entrées des cpteurs ussi bien que dns l odélistion et dns les entrées des propriétés des tériux ssociés. Pr conséquence, il existe un besoin inhérent pour que les éléents du systèe de pronostic soient à bse létoire. Étnt donné que l odélistion stochstique considère quelques prètres du systèe coe létoires, lors l loi de propgtion de Pris devient stochstique. Les données de dignostic perettent de prendre l longueur initile de fissure en tnt qu'une preière vrible létoire et l contrinte de chrgeent en tnt qu'une seconde vrible létoire. otre odèle de dégrdtion stochstique est donné sous l fore d'une reltion ~ récursive relint deux rélistions consécutives de dégrdtion D ( ~ ) et D ~ ( ~ ) en deux cycles voisins vec un incréent de doge d D ~ à l fin de chque cycle de chrgeent. otons que chque rélistion de dégrdtion est fonction d'une rélistion de
longueur de fissure ~ donnée à son tour en fonction d'une longueur initile de fissure ~ rendue létoire. Donc, l reltion récursive du odèle décrit l'évolution de l dégrdtion D ~ en fonction des vribles létoires suivntes: longueur initile de fissure ~, chrgeent ~ et l longueur cournte de fissure ~. A chque cycle de chrgeent ( ), l'indice de dégrdtion D ugente d'une quntité dd prtnt de D jusqu'à une vleur unitire (D ) qui n'est utre que l'étt de défillnce du systèe. Ainsi, le odèle de pronostic devient plus précis dns l prédiction des RUL. Les résultts des durées de vie résiduelles sont obtenus pour le doge dns les cs linéires et non-linéires et les différences sont justifiées ussi pr les tendnces ultiples de dégrdtion. L propgtion stochstique de fissures iplique des odèles vec des prètres létoires qui peuvent être estiés en utilisnt les siultions de Monte-rlo. es prètres stochstiques sont ffectés pr certines probbilités de rélistion influnt les RUL résultntes déduites des trjectoires de dégrdtion. Encore une fois, les deux êes pplictions déjà tritées concernnt les suspensions et les pipelines sont considérées de nouveu dns ce qutrièe chpitre. oe perspectives, il est plnifié de ieux développer l éthodologie de pronostic proposée et l'ppliquer sur un lrge enseble des systèes dyniques. eci est rélisé en prennt en considértion d'utres lois nlytiques de l propgtion de fissures et d'utres lois de cuul de doge. Ajoutons sur ceci qu'un plus grnd nobre de prètres de bse peuvent être ssiilées coe vribles létoires, à noter, les prètres des tériux et de l'environneent et d'utres prètres géoétriques et écniques. De êe, des nouvelles lois probbilistes utres que l loi orle et l loi Log-orle peuvent être explorées. Aussi, il est plnifié de ieux border l vribilité des durées de vie stochstiques et d'en déduire un fisceu des courbes de dégrdtion. En effet, des prètres de bse rendus ~ létoires boutissent à une trjectoire de dégrdtion probbilisée D ( ). Ainsi, un fisceu
de courbes D() est obtenu pour lequel une courbe oyenne et une courbe d'écrt-type sont déduites. Pr conséquence, une courbe crctéristique D K () peut être clculée en teres d'un frctile α% qui dépend du niveu cceptble du risque. L vleur crctéristique de RUL est donc déduite à prtir de l courbe D K ().