ESPOO 2005 VTT WORKING PAPERS 35 Fatgue analyss framewor to support fleet management fatgue damage data from montored structures Devaton crtera for Fatgue Lfe Fatgue Lfe comparson Safe Lfe crtera Next flght operatons Bayesan networ for fatgue data Crtera for extremalty of data Data outler assessment Safety evaluaton Varaton crtera Flght tas performance assessment Avalablty crtera Fleet avalablty assessment Cost crtera Mantenance cost assessment Objectves of flght operatons Plannng of flght tas mxes / flght operatons Fatgue Analyss for Fleet Management Usng Bayesan Networs Tony Rosqvst VTT Industral Systems
ISBN 951 38 6582 7 (URL: http://www.vtt.f/nf/pdf/) ISSN 1459 7683 (URL: http://www.vtt.f/nf/pdf/) Copyrght VTT 2005 JULKAISIJA UTGIVARE PUBLISHER VTT, Vuormehente 5, PL 2000, 02044 VTT puh. vahde 020 722 111, fas 020 722 4374 VTT, Bergsmansvägen 5, PB 2000, 02044 VTT tel. växel 020 722 111, fax 020 722 4374 VTT Techncal Research Centre of Fnland, Vuormehente 5, P.O.Box 2000, FI 02044 VTT, Fnland phone nternat. +358 20 722 111, fax +358 20 722 4374 VTT Tuotteet ja tuotanto, Otaaar 7 B, PL 1705, 02044 VTT puh. vahde 020 722 111, fas 020 722 7076 VTT Industrella system, Otsvängen 7 B, PB 1705, 02044 VTT tel. växel 020 722 111, fax 020 722 7076 VTT Industral Systems, Otaaar 7 B, P.O.Box 1705, FI 02044 VTT, Fnland phone nternat. +358 20 722 111, fax +358 20 722 7076 Techncal edtng Marja Kettunen
Publshed by Author(s) Rosqvst, Tony Seres ttle, number and report code of publcaton VTT Worng Papers 35 VTT WORK 35 Ttle Fatgue Analyss for Fleet Management Usng Bayesan Networs Abstract A fatgue analyss framewor for estmatng and predctng (cumulatve) fatgue damage of arcraft structures, based on fatgue damage data from a fleet of arcraft, s descrbed. The data model s defned by a Bayesan Networ. Assessments related to safety, mantenance and lfe cycle cost, flght tas performance and fatgue damage dstrbuton between structures, are supported. The fatgue analyss framewor gudes the flght planner n specfyng flght syllab that maxmse operatonal and tranng effectveness whle smultaneously meetng safety and cost constrants. The mplementaton of the fatgue analyss framewor requres computersed functons for data sortng, algorthms related to the Bayesan statstcal nferences of the fatgue damage data model, and a graphcal dsplay of multple analyss results. Keywords fleet management, fatgue management, fatgue lfe, fatgue load, fatgue damage analyss, Bayesan Networ Actvty unt VTT Industral Systems, Otaaar 7 B, P.O.Box 1705, FI 02044 VTT, Fnland ISBN Project number 951 38 6582 7 (URL: http://www.vtt.f/nf/pdf/) G4SU00326 Date Language Pages November 2005 Englsh 25 p. Name of project Seres ttle and ISSN VTT Worng Papers 1459 7683 (URL: http://www.vtt.f/nf/pdf/) Commssoned by Fnnsh Ar Force Headquarters Publsher VTT Informaton Servce P.O. Box 2000, FI 02044 VTT, Fnland Phone nternat. +358 20 722 4404 Fax +358 20 722 4374 3
Foreword The objectve of the worng report s to outlne a framewor for fatgue lfe analyss and predcton to support the fleet management of FAF arcraft. The framewor s based on Bayesan Networ modellng. The research has been conducted durng 2004 n cooperaton between Asla Sljander (Group Manager), Saul Luonen, Ma Bäcström, Kejo Kos and Tony Rosqvst. Research Professor Urho Pulnen provded comments and feedbac to the development of the basc data model. The author wants express hs grattude to the FAF HQ for fnancng the research. It s the hope of the author that the worng report wll rase comments wthn the communty of fleet managers controllng the usage of FAF arcrafts and fatgue experts n general. It s also the hope of the author that the worng report wll support and gude the preparaton of future project proposals amed at developng fleet and fatgue management of the FAF arcraft. Espoo 1.6.2005 Tony Rosqvst 4
Contents Foreword...4 1. Introducton...6 2. Fatgue damage analyss framewor to support fleet management...8 2.1 Nomenclature*...8 2.2 Descrpton of the fatgue analyss framewor...10 2.2.1 Basc decson analyses for fleet management...10 2.2.2 Man assumpton for modellng fatgue damage...12 2.2.3 The nferental procedure for estmaton and predcton...13 2.3 Bayesan Networ model for fatgue damage data...14 2.3.1 Basc nferences and predctons...14 2.3.2 Data outler ndcaton...17 2.3.3 Fatgue Lfe comparson...18 2.3.4 Safety evaluaton...18 2.3.5 Flght tas performance assessment...19 2.3.6 Fleet avalablty assessment...20 2.3.7 Mantenance cost assessment...20 2.3.8 Syllab plannng...21 2.4 Refnements of the Bayesan Networ model...22 2.4.1 Refnement of the model by uncouplng planes and plots...22 2.4.2 Value of nformaton...22 3. Software mplementaton of the fatgue analyss framewor...23 4. Conclusons...24 References...25 5
1. Introducton The motvatons for the worng report are the followng statements n the VTT report BVAL33-011139 / AOS (Sljander, 2001): Fatgue management covers the organsaton and management of any functons requred to eep the FAF arcraft fleet flyng untl the planned Out of Servce Date wthout the need for major modfcatons or repars to the arcraft s load carryng structure, whle enablng the operatonal and tranng objectves to be met smultaneously precludng any ssues relatng to structural fatgue to affect [compromse] the safety of flght, arcraft avalablty or costs of operaton. (secton 2, p. 5). The goal for the fatgue management of the FAF can currently be formulated as follows: Adjust the content of tranng syllab and the peacetme operatonal usage such that operatonal and tranng effectveness s maxmsed whle smultaneously meetng a gven Out of Servce Date (ch. 3, p. 5). The am of the worng report s to formulate more precsely the fatgue management challenges stated above. The scope s lmted to problem formulaton and a mathematcal outlne of a fatgue analyss framewor supportng fatgue management of a fleet of arcraft. Possble operatve constrants n mplementng the analyss framewor n the currently adopted fatgue management functons are not addressed. The man challenge for the flght syllab planner s to specfy sets of flght tass for each arcraft such that the flght syllab maxmses the value of the tranng for the fleet whle complyng wth safety and cost crtera. A specfc challenge s to specfy flght syllab such that the lfe cycle cost constrant of the fleet s met when the planned Out of Servce date s reached. The arcraft perform dfferent sets of flght tass.e. flght syllab, where a flght tas can be vewed as the basc fatgue load unt or buldng bloc n the syllab. A flght syllabus s a set of flght tass wth a specfc tranng objectve. When a fleet performs a gven flght tranng program, the partcpatng planes usually perform dfferent flght syllab n order for the fleet to fulfl the overall objectve of the tranng program. The structures of the arcraft are therefore subject to dfferent numbers of flght tass, each wth ts own fatgue load and damage characterstc. From the pont of vew of fleet management t s desrable to get a predcton whether any structure of the arcraft wll exceed a predefned level of crtcal fatgue damage wth some probablty durng the next flght syllabus (set of flght tass). The crtcalty 6
can be evaluated wth respect to safety, NDT testng requrement, replacement need of structure, mantenance cost of flght syllab, etc. The fatgue damage related to a flght tas s not exactly estmable / predctable. For smlar flght tass, the fatgue load and damage on a structure wll vary due to varablty n plot performance and external condtons. The uncertanty related to the fatgue damage of a structure, can, however, be quantfed by collectng data n the form of {flght tas d, fatgue damage measurement value} and mang statstcal nference on fatgue damage parameters of a data model. The probablstc nference on data model parameters s based on stran gauge-.e. OLM-measurements, and acceleraton sensor-.e. g-measurements depctng fatgue load of a structure. These measurement scales are functonally related to a fatgue ndex, such as FI or FLE, depctng fatgue damage. It s generally consdered that OLMmeasurements better capture the structural damage caused by the load, compared to g-measurements, for some structures. OLM measurement systems are, however, more expensve and usually have to be retroftted to the structures. OLM-measurement values are obtaned from a subpopulaton of structures whereas g- measurement values are obtaned from all structures n the fleet. Obvously, t s preferable to use all nformaton (evdence) that s avalable for the nference. An nferental procedure that lns dfferent types of evdence s provded by Bayesan statstcal theory (Gelman et al., 2004). The nferental structure s depcted by Bayesan Networs. The Bayesan approach s therefore adopted n the development the fatgue analyss framewor ntroduced n the worng report. 7
2. Fatgue damage analyss framewor to support fleet management Parameter Flght tas Defnton 2.1 Nomenclature* A defned rule for performng a flght, usually ndcated by syllabus code and msson type. Flght syllabus Set of flght tass for a sngle arcraft wth specfc tranng objectves. Flght syllab Set of flght tass assgned to several arcraft wth specfc tranng objectves of the fleet. =1,, N Flght tas ndex. ~ j = 1,..., L,... L Index of occurrence of flght tas. L denotes the performed number of flght tas n the fleet. L ~ denotes the number of flght tas planned for the fleet. = 1,,K Index of structure unt or arcraft ( talnumber ) ~ j = 1,..., L,..., L Index of the occurrence of flght tas related to structure unt. denotes the past number of flght tas, whereas L ~ m(olm) m(g) planned number of flght tas for structure unt. We have the relatonshps L = K = 1 K ~ ~ L and L = L. = 1 L denotes the Measurement sgnal obtaned from the OLM-system wth stran gauges. The sgnal represents fatgue load of the montored structure (Fg. 1a). Measurement sgnal obtaned from the g-measurement system wth acceleraton sensors. The sgnal represents fatgue load of the montored structure (Fg. 1a). x Indrect measure x = f ( m( OLM )) of fatgue damage. Functon f s an ncreasng monotonous functon (Fg. 1a). y Indrect measure y = h( m( g)) of fatgue damage. Functon h s an ncreasng monotonous functon (Fg. 1a). µ Mean of fatgue damage related to flght tas. σ Standard error of fatgue damage related to flght tas. 8
η y Bas of measurement value y. σ y Standard error of measurement value y. σ x Standard error of measurement value x. θ j ψ ~ ψ H H H ~ H ~ Z = (z 1,.,z,.,z N ) Z = (z 1,...,z N ) Fatgue damage of structure unt nduced by the j th flght of flght tas. Cumulatve fatgue damage or fatgue lfe of structure unt, also denoted by fatgue ndex FI (reference levels are defned n Fg. 1b). Cumulatve fatgue damage of structure unt after planned set of flght tass, also denoted by F I ~ (reference levels are defned n Fg. 1b). Matrx of the number of dfferent flght tass performed nvolvng all smlar structures of the fleet (see text). Array of the number of dfferent flght tass performed nvolvng structure unt (see text). Matrx of the number of dfferent flght tass planned for the fleet (see text). Matrx of the number of dfferent flght tass planned to nvolve structure unt (see text). Data matrx of fatgue damage measurement value pars z = ((x 1, y 1 ),, (x j, y j ), ) from the whole fleet lsted accordng to flght tas Data matrx of fatgue damage measurement value pars z = ((x 1, y 1 ),, (x j, y j ), ) of structure unt (Z Z) *the most mportant quanttes n the fatgue analyss framewor are lsted Measures related to a flght tas : fatgue load m(g) h fatgue damage y m(olm) f x Fgure 1a. Measures (data) related to the fatgue of an arcraft structure subject to flght tass. 9
fatgue brea pont ψ B crtcal fatgue level ψ L NDT trgger level ψ NDT fatgue baselne (at tme pont zero): ψ 0 Fgure 1b. Cumulatve fatgue damage (fatgue lfe) reference levels used to defne probablstc decson crtera used n the assessment and evaluaton of the condton of an arcraft structure. 2.2 Descrpton of the fatgue analyss framewor 2.2.1 Basc decson analyses for fleet management In the followng, we treat an arcraft structure subject to smlar flght tass as a statstcal unt. A statstcal populaton s the group of smlar arcraft structures subject to smlar flght tass. Varatons n the observed fatgue damage measurements are due to varatons n plane&plot combnatons and human&machne nteractons (plot performance), as well as measurement errors. As there are no trac record on the plane and plot combnatons, t s not possble by statstcal means to compare the fatgue damage contrbutons of the plots; f an extreme fatgue measurement value s observed, we need separate analyses to dentfy the causes. As a consequence, we hold all measurement values obtaned from the plot & plane unts statstcally equally representatve for the consdered flght tas. Due to the probablstc approach, decson-mang s guded by defnng probablstc decson crtera (rs crtera) representng fleet management strategy. Rs crtera can be defned to support ) data outler ndcaton to attract attenton to possble extreme fatgue damages or faults n the measurement system; ) fatgue lfe comparson to balance the fatgue loadng / damage of structures n the fleet, 10
) v) safety evaluaton to chec whether rs lmts are compled wth or not; flght tas performance assessment to support specfcaton of operatonal requrements related to the executon of a flght tas; v) fleet avalablty assessment to chec the avalablty of the requred number of arcraft for a planned flght operaton; v) mantenance cost assessment to chec f budget and Lfe Cycle Cost constrants are met. The above assessments and evaluatons may be performed after every OLM- and g- measurement. As the fatgue damage data hstory grows, refned estmatons of the fatgue damages related to flght tass, and predctons of the cumulatve fatgue damage related to the planned flght syllab, can be made. The assessments and evaluatons may ndcate a need for re-plannng the flght syllab. The teratve nature of fatgue load and damage measurement, fatgue damage estmaton, fatgue lfe predcton, and syllab plannng, wthn the fatgue analyss framewor, s shown n Fg. 2. The analyss framewor can be vewed as an emprcal fatgue analyss framewor supportng what-f analyss of optonal flght syllab and tranng programs. As the plannng of the flght syllab s a complex tas of talorng together sets of flght tass for each arcraft, wth several operatonal and tranng objectves to acheve together wth safety and cost constrants, plannng cannot, n general, be reduced to an optmsaton problem yeldng an optmal assgnment of flght tass and syllab for plot & plane - unts untl the Out of Servce Date. 11
Fatgue analyss framewor to support fleet management fatgue damage data from montored structures Devaton crtera for Fatgue Lfe Fatgue Lfe comparson Safe Lfe crtera Next flght operatons Bayesan networ for fatgue data Crtera for extremalty of data Data outler assessment Safety evaluaton Varaton crtera Flght tas performance assessment Avalablty crtera Fleet avalablty assessment Cost crtera Mantenance cost assessment Objectves of flght operatons Plannng of flght tas mxes / flght operatons Fgure 2. Analyss framewor for gudng the plannng of flght syllab teratvely as observatons on fatgue damage become avalable. 2.2.2 Man assumpton for modellng fatgue damage The man assumpton for modellng fatgue damage related to a flght tas, and the accumulaton of fatgue damage as a functon of flght tass, s order-nvarance,.e. smlar flght tass damage a structure statstcally smlarly rrespectve of the order and dstrbuton of the flght tass n a flght syllab. As a frst corollary, the overall damage ncurred by a flght syllabus (comprsng of smlar and dfferent flght tass) s statstcally the same regardless of the order of the flght tass n the syllabus. As a second corollary, varatons n the external envronment and the man-machne nteractons (plot performance) are assumed to be jontly statstcally stable (wth no trend n tme). As a consequence of the fatgue damage order-nvarance assumpton, we can model fatgue damage accumulaton by an addtve fatgue damage functon (see secton 2.3.1). 12
2.2.3 The nferental procedure for estmaton and predcton The nferental procedure s two-staged: Frstly, all fatgue measurement values obtaned from all structures n the fleet, and related to smlar structures and flght tass, are used for statstcal nference on parameters denotng fatgue damage. Secondly, the obtaned probablty functons on the fatgue damage parameters are used for estmaton and predcton of the cumulatve fatgue damage (=fatgue lfe) of a structure unt (arcraft level). The nformaton flow between fatgue measurement, nference, estmaton, predcton, and assessment, n the fatgue analyss framewor, s outlned below: Data and nference at fleet level (nference step 1): Flght tas hstory of fleet H = ( L,..., L,..., L )) 1 N Vector denotng the flght tas hstory of the fleet as the number of flghts dstrbuted over the flght tass,.e. the total fatgue load n terms of flght tas type related to an assamble of structures of some type. Fatgue damage measures from fleet z. Z = z. z = (( x = (( x 1 = (( x, y, y 1 1, y ),...,( x ),...,( x 1 ),...,( x, y 1 11 11 1L1 1L1 N L L, y N L, y )) )) )) N Data matrx representng the fatgue damage data: (x, y) = (f(m(olm), g(m(g)) data pars. Note that for most data entres x = N/A as OLM-measurements are only avalable from a few structures! L Inference on data model parameters gven fleet data pdf(fi) FI Set of probablty functons related to the data model parameters. The prmary parameters are the fatgue damage parameters related to the dfferent flght tass and the cumulatve fatgue lfe FI. Predcton and what-f analyss at arcraft level (nference step 2): Fatgue loadng of a structure Flght syllabus hstory: H = ( L,..., L,..., L )) 1 Planned ~ flght syllabus: H = ~ ~ ~ ( L,..., L,..., L ) 1 N N Estmaton and predcton of fatgue lfe of structure unt p(fi ) FI Evaluaton of complance wth rs crtera p(fi ) FI Matrx denotng planned flght syllabus,.e. the number of dfferent flght tass that structure unt wll be subject to. Probablty functon of fatgue lfe of structure unt subject to the fatgue loads of the performed (planned) flght tass Are structure-specfc rs crtera compled wth or not? If yes, then no extra acton. If no, perform requred acton. 13
In the the second step, estmatons and predctons on the fatgue lfe of a structure unt wll be computed and used n the evaluaton of complance wth gven rs crtera for decson-mang regardng the feasblty and acceptablty of the planned flght syllabus of each arcraft. The outcome of the evaluaton gudes the flght syllabus planner n a manner correspondng to what-f analyses. 2.3 Bayesan Networ model for fatgue damage data Reference to Fg. 2: 2.3.1 Basc nferences and predctons data sortng crtera fatgue damage data Bayesan Networ for fleet data Basc nferences and predctons 1. Each plane has a number of dfferent structures (wng, hull, etc) wch are montored by a set of sensors. In the followng, one structure type wll be consdered wthout any loss of generalty of the approach. 2. Each structure s subject to fatgue loads determned by the flght tass related to a flght syllabus. The dfferent flght tass = 1,, N occur n any order and number n dfferent flght syllab. The content of the flght syllab s determned by the tranng objectves. 3. The order of fatgue loads n the fleet, related to flght tas, s ndcated by j = 1,, L, where L s the current accumulated number of performed flght tas. The order of fatgue loads from flght tas, experenced by the structure unt, s ndcated by j = 1,, L. 4. Each performed flght tas s coupled wth a g-measurement value ndcatng fatgue load of a structure. Each performed flght tas s also coupled wth an OLMmeasurement value ndcatng fatgue load of a structure that belongs to the subpopulaton of structures (.e. fleet leaders) that are OLM-montored. 5. Each structure goes through a set of flght tass before the next fatgue load and fatgue damage analyss. 14
6. Based on the basc assumptons stated n secton 2.1., an addtve fatgue damage functon defnng the fatgue lfe (cumulatve fatgue damage) of a structure unt can be defned as: N L ψ = θ (1) = 1 j= 1 j where ψ denotes the fatgue lfe of a structure unt, θ j denotes the underlyng (hdden) fatgue damage related to the j th flght of flght tas, and L s the number of flght tass performed nvolvng structure unt. The quantty ψ wll also denote the fatgue ndex FI. 7. In predcton the above equaton has to be extended to ncorporate the planned set of future flght tass (planned flght syllabus): ~ N L + L ~ ψ = θ (2) = 1 j= 1 j where ~ ψ denotes the predctve cumulatve fatgue damage of a structure. L ~ denotes the number of planned flghts of flght tass. The quantty ~ ψ also denotes the predcted fatgue ndex F I ~. 8. In the Bayesan statstcal approach dfferent types of evdence can be combned for the nference on the underlyng fatgue damage parameters θ j. We use the x = f(m(olm)) and the y = h(m(g)) measures as evdence on fatgue damage and use them n the nference on the fatgue damage parametes θ, ψ and ~ ψ. 9. The dependence between the fatgue damage parameters n the data model, and the (x,y)-measurement values related to flght tas, are depcted by the Bayesan Networ n Fg. 3. For each flght tas = 1,, N, a smlar Bayesan Networ s specfed. (For those structures that are not OLM-montored we mar the x-data entry by N/A); 10. The Bayes Networ n Fg. 3 ncludes, n the frst nferental step, all evdence obtaned from the whole populaton of the montored and smlar structures (wng, helm, etc.). In the second nferental step, the fatgue lfe of each structure unt s estmated and/or predcted (see secton 2.2.3). j 15
2 2 [ µ σ ] N( µ 0, σ / κ 0 ) 2 2 2 [ σ ] Inv χ ( υ, σ ) 0 0 2 [ logθ µ, σ ] N( µ, σ ) j j = 1,..., L 2 [ logxj θ j, σ x ] N( θj, σ x ) 2 [ log y θ, η, σ ] N( θ + η, σ ) j j j = 1,..., L y y j y y ( θ 1,.., θ ) ( x 1,..., x ) ( y 1,..., y ) L L L 2 [ logθ µ, σ ] N( µ, σ ) j ~ j = 1,..., L,..., L ψ = N L = 1 j= 1 ( θ 1,..,θ ) ( θ,.., ) 1 θ ~, L + L θ j ψ L ~ N L + L ~ ψ = θ = 1 j j= 1 ψ ~ 2 2 2 [ σ ] χ ( υ ) x Inv σ x 0, x 0 ( 0,σ x ) ( η, ) y σ y 2 2 [ ηy σ y ] N( ηy, σ / 0 ) 0 y κ 0 2 2 2 [ σ ] Inv χ ( υ, σ ) y y0 y0 Fgure 3. A Bayesan Networ depctng the relatonshp between the fleet data L z ( ) = x j, y j, the hdden fatgue damage parameters θ j and ther dstrbuton j= 1 parameters ( µ, σ ), the measurement error dstrbuton parameters (, σ ) and ( η, σ ) 0, and the estmated and predcted fatgue lfe parameters x y y ψ, ~ ψ of structure unt. Index denotes flght tas, ndex j the order of flght, and the structure unt. 11. Features of the fatgue damage Bayes Networ :! The varaton n the performance of the plot, the combnaton of plot and plane, the external condtons when performng a specfed flght tas s reflected n varatons n the fatgue load and fatgue damage and s represented by the probablty functon parameters ( ) σ matrx Z. µ,, specfed by fleet fatgue damage! It s assumed that the measurement values x = f(m(olm)) are unbased, whereas y = h(m(g)) are based, wth the log-transformed values normally dstrbuted gven the parameters ( 0, σ x ), ( µ y, σ y ). Thus the measurement error related to the OLM-measurement from fleet leaders has zero mean. 16
! The pror probablty functons related to the parameter pars ( µ, σ ) and (, σ ), ( µ, σ ) 0 may be defned accordng to Gelman et al. (2004). Other x y alternatves are possble. y! In the nference depcted by the Bayes Networ we obtan the (predctve) posteror probablty functons related to the fatgue damage parameters gven the fatgue damage data of the fleet,.e. ~ { p( θ j z ), j = 1,..., L,..., L + L, = 1,..., N}. These probablty functons are then used to compute the fatgue lfe estmates and predctons used n the varous assessments after the data-analyss, as shown n Fg. 2.! The nference s n practce obtaned numercally by Marov Chan Monte Carlo methods (Gelman et al., 2004) f analytc solutons are not dervable. 2.3.2 Data outler ndcaton Reference to Fg. 2: Crtera for extremalty of data Basc nferences and predctons Data outler ndcaton Data acceptance / rejecton Data analyss / fleet management queston: What are the reasons for extreme fatgue damage measurement value(s)? The analyss procedure s as follows: 12. The extremalty of an observaton of fatgue damage; ( x j, y j ) can be assessed n terms of ts lelhood. Denote a p-fractle related to e.g. y-data by y p. Let the probabltes p = p l and p = p u defne fractles related to extremely small and bg fatgue damage observatons, respectvely. The probabltes of observng extremal y measurement values related to the j th realsaton of flght tas are P( Yj > y p θ j, ηy, σ y ) and P( Yj < y p θ j, ηy, σ y ). By specfyng the probablty lmt u l values p l and p u, the correspondng p-fractles y and y can be determned. p l p u 13. Extremal fatgue damage y-observatons, low or hgh, related to flght tas I, can l u y y y > y, respectvely. now be alerted by the ndcator functons ( < ) and ( ) j p l j p u 17
2.3.3 Fatgue Lfe comparson Reference to Fg. 2: Devaton crtera for fatgue lfe Basc nferences and predctons Fatgue lfe comparson Fatgue lfe dscrepancy ranng / evaluaton Fleet management queston: Is the populaton of smlar structures n the fleet loaded evenly? The analyss procedure s as follows: 14. The estmated structure specfc fatgue lves { ψ } can be compared wth each other n terms of a devaton measure, such as D = ψ ψ (where ψ s the expected sum of the random varablesψ, = 1,,K), to chec for balanced fatgue loadng of the structures. 15. A probablstc crteron ndcatng unbalanced fatgue damage can be defned as C P( D d Z ), = 1,..., K that s the posteror probablty of observng a devaton larger than some crtcal value d C. 16. Crtcal dfferences n fatgue damages can be alerted by an ndcator functon D C P D d Z α, = 1,...,, ( ( ) ) K where α s a pre-defned rs level (α << 1). Reference to Fg. 2: 2.3.4 Safety evaluaton Rs crtera / safe lfe crtera Basc nferences and predctons Safety evaluaton Safety evaluaton Fleet management queston: Are the next planned flght tass safe enough? 18
The analyss procedure s as follows: 17. As the cumulatve fatgue damage ψ of structure unt ncreases t approaches the crtcal fatgue level ψ L specfyng the "safe lfe" of the structure. (Ths lmt should be selected such that the probablty of fndng a macro crac n the structure s mnscule when ths lmt s reached.) 18. The predctve posteror probablty of structure unt exceedng the crtcal fatgue lmt ψ L ~ ~ ~, before the end of the planned flght mxes ( L1,..., L,..., LN ), s ~ ~ ~ ~ L P Ψ L,..., L,..., L ψ Z, 1,..., ( ( ) ) K 1 N = 19. A rs ndcator functon alertng of a cumulatve fatgue damage predcton exceedng the crtcal fatgue lmt ψ L by a probablty larger than a pre-defned rs S ~ L P Ψ ψ Z α level α << 1, can be wrtten as ( ( ) ) Reference to Fg. 2: 2.3.5 Flght tas performance assessment Crtera for varaton Basc nferences and predctons Flght tas performance assessment Flght tas dscrepany ranng / evaluaton Fleet management questons: Is the fatgue damage related to any flght tas wdely spread? ; Are the techncal specfcatons related to a flght tas too broadly stated, leavng room for subjectve nterpretaton about how to perform the flght tas? The analyss procedure s as follows: 20. A measure that taes nto account both the expected value and the standard devaton of the fatgue damage related to flght tas s the posteror coeffcent of CV z = E σ z / E µ z. varaton: ( ) [ ] [ ] 21. Those flght tass = 1,,N, whch are assocated wth hgh CV - values, are more uncertan and may be prortzed for further fatgue damage analyses where the causes of the varatons are dentfed, supportng better specfcaton of the operatonal requrements related to the flght tass. 22. Probablstc crtera can be defned for alertng excessve varaton n flght performance. 19
2.3.6 Fleet avalablty assessment Reference to Fg. 2: Crtera for fleet avalablty Basc nferences and predctons Fleet avalablty assessment Fleet avalablty for planned flght operatons Fleet management queston: Are the fatgue damage levels of structures such that enough structures are avalable for the next planned flght program? The analyss procedure s as follows: 23. The predctve densty of the cumulatve fatgue damage for a structure depends on the ~ ~ ~ ~ p Ψ L,..., L,..., L Z, 1,..., planned flght tas mx for the structure: ( ( ) ) K 1 N = 24. If K structures (arcraft) are requred to meet the operatonal objectves set for a flght program then unavalablty of enough safe structures (at rs level α) can be ndcated K based on the safety ndcator S A S ~ L : ( P( Ψ ψ Z ) α ) K K' = 1 25. If unavalablty s ndcated, alternatve flght tas mxes / flght programs need to be desgned or structures have to be renewed. 2.3.7 Mantenance cost assessment Reference to Fg. 2: Cost Crtera NDT-test crtera Lfe Cycle Cost crtera Replacement Cost crtera Basc nferences and predctons Mantenance cost assessment Mantenance cost acceptance / rejecton Fleet management queston: Do the next planned flght tass meet mantenance cost crtera? 20
The analyss procedure s as follows: 26. The utlsed mantenance strategy s condton-based replacement,.e. when a structure s fatgue lfe exceeds, for nstance, the crtcal fatgue lfe ψ L, the structure s consdered used, and s replaced by a smlar new structure, ncurrng a cost ρ. 27. The expected number of replacements of smlar structures after the next planned fleet K K R operaton s { > } = ~ L nr = E ~ L ψ ψ P( Ψ > ψ Z ), where R (.) s an ndcator functon = 1 = 1 ndcatng that the crtcal fatgue damage lmt (safe lfe) s exceeded. 28. The expected cost of replacements s now c R = ρ R * nr. 29. Every planned flght program can be assocated wth an expected number of NDT-tests to be performed for the K structures of smlar type. Now, the expected number of NDT-tests K K NDT s { > } = ~ NDT nndt = E ~ NDT ψ ψ P( Ψ > ψ Z ), where the ndcator functon = 1 = 1 ndcates that a NDT-test, wth a cost, s performed when the fatgue damage treshold value ψ NDT s exceeded. ρ NDT 30. The expected NDT-cost assocated wth the flght program s c NDT = ρ NDT * nndt 31. If cost crtera have been set for NDT- and replacement costs, then complance / noncomplance can be checed for, steerng the plannng of flght programs. 32. The cost assessment can be extended to Lfe Cycle Cost assessment by assesng the above costs for flght programs extendng to the planned Out of Servce Date of the fleet. R 2.3.8 Syllab plannng Objectves of flght operatons Assessments related to fatge damage Plannng of flght tas mxes / flght operatons Next flght operatons 33. The plannng of flght syllab has to meet several objectves related to tranng, safety and cost. Safety and cost objectves may be looed as constrants for the specfcaton of flght syllab. In partcular, the planned lfetme of the fleet,.e. the 21
Out Of Servce date, has to be met. Utlsng the fatgue analyss framewor, plannng s guded by what-f analyses of fatgue lfe predctons related to flght operatons, and reles on the slls of the planner to specfy flght syllab that are feasble. 2.4 Refnements of the Bayesan Networ model 2.4.1 Refnement of the model by uncouplng planes and plots Ideally, plots and planes would be systematcally mxed after each flght syllabus. A personal trac record of the fatgue damage ncurred by each plot would serve as a bass for statstcal analyss of devatng plot performances for partcular flght tass under smlar condtons. Such nformaton could be used to montor and learn from those plots that acheve the objectves of the flght syllabus wth mnmal fatgue lfe expendture. 2.4.2 Value of nformaton The fatgue data model can, n prncple, be used to assess the lmt value of addtonal nformaton by nstallng more OLM- measurement systems n the populaton of arcraft. By more accurate estmatons and predctons t s possble to resolve uncertanty and therefore prevent premature deleton of structures based on the adopted rs crtera. 22
3. Software mplementaton of the fatgue analyss framewor The mplementaton of the analyss framewor outlned above for fatgue management of a fleet of arcraft requres the development of a software tool. The artefacts of the software development process should be: ) a data sortng nterface for groupng fatgue load sets whch are used as nput for the basc nferences n the fatgue analyss framewor; ) ) v) an algorthm that computes the nferences needed for the dfferent subassessments and evaluatons; a graphcal nterface for smultaneous dsplay of sub-assessment results; a verfcaton report on the proper computer mplementaton of the Bayesan data model by comparng test data wth analytcally obtaned results; v) a valdaton report based on real fatgue damage measurement values and observatons of macro cracs n structures The resources needed for the development of the framewor s estmated to be 1,5 2 man-years. 23
4. Conclusons A fatgue analyss framewor for estmatng and predctng (cumulatve) fatgue damage of arcraft structures, based on fatgue damage data from a fleet of arcraft, s descrbed. The data model s defned by a Bayesan Networ. Assessments related to safety, mantenance and lfe cycle cost, flght tas performance and fatgue damage dstrbuton between structures, are supported. The fatgue analyss framewor gudes the flght planner n specfyng flght syllab that maxmse operatonal and tranng effectveness whle smultaneously meetng safety and cost constrants. The mplementaton of the fatgue analyss framewor requres computersed functons for data sortng, algorthms related to the Bayesan statstcal nferences of the fatgue damage data model, and a graphcal dsplay of multple analyss results. 24
References Sljander, A. A revew of recent aeronautcal fatgue nvestgatons n Fnland untl March 2001, Research report BVAL33-011139 / AOS, VTT Research Centre of Fnland, Espoo, 2001. Gelman, A., Carln, J., Stern, H. Rubn, D. Bayesan Data Analyss (2 nd ed.), Chapman & Hall, 2004. Keeney, R., Raffa, H. Decsons wth Multple Objectves : Preferences and Value Trade-offs, Cambrdge Unversty Press, 1993. 25
VTT WORKING PAPERS VTT TUOTTEET JA TUOTANTO VTT INDUSTRIELLA SYSTEM VTT INDUSTRIAL SYSTEMS 13 Pötry, Jyr & Hänen, Ka. Laatuvrheet alhanntaonepajossa havantoja vuonna 2004. Laatual-projetn osaraportt. 2005. 41 s. 23 Salonen, Tapo & Sääs, Juha. Tuotetetostandarden äyttö tuotannossa. 2005. 19 s. 28 Sääs, Juha, Salonen, Tapo & Paro, Jua. Integraton of CAD, CAM and NC wth Step-NC. 2005. 23 p. 35 Rosqvst, Tony. Fatgue Analyss for Fleet Management usng Bayesan Networs. 2005. 25 p. ISBN 951 38 6582 7 (URL: http://www.vtt.f/nf/pdf/) ISSN 1459 7683 (URL: http://www.vtt.f/nf/pdf/)