An introduction to Reliability Analysis

Transcription

1 An ntroducton to Relablty Analyss Vncent DENOEL Unversty of Lege, ArGEnCo, MS 2 F Departement of Archtecture, Geology, Envronment and Constructon - Sold, Structures and Flud Mechancs Dvson - January 2007 Ths redacton of ths document and the development of the llustratons could be realzed thanks to Prof. H. KUSAMA, School of Desgn and Archtecture, Unversty of Nagoya Cty, Japan. The author s grateful to the Unversty of Nagoya Cty, ts Unversty Board of Drectors and the concerned Faculty and Department Meetngs. The redacton of ths orgnal document (text and fgures) has been completed between November 23 rd, 2006 and January 25 th, 2007, durng an nvted stay of the author at the Unversty of Nagoya Cty. It s warmly acknowledged for ths nvtaton.

2 Contents 1 Introducton 3 2 Relablty analyss FrstOrderSecondMoment(FOSM) AdvancedFrstOrderSecondMoment(AFOSM) Presentatonofthemethod LnkbetweentheFOSMandAFOSMmethods Detalsoftheresoluton AdvancedFrstOrderSecondMomentforCorrelatedvarables(AFOSMC) Second-OrderRelabltyMethods(SORM) Frst-OrderGaussanSecondMomentMethod(FOGSM) Frst-OrderGaussanApproxmatonMethod(FOGAM) Summary Illustratons Bendngmodel(uncorrelatedvarables) Probablstcanalyss:analytcalapproach Relabltyanalyss:analytcalapproach Relablty analyss: llustraton of the numercal resoluton (FOSM) Relablty analyss: llustraton of the numercal resoluton (AFOSM) Relablty analyss: llustraton of the nvarance prncple (FOSM.vs. AFOSM) Relabltyanalyss:4-varableproblem Relabltyanalyss:exampleofdvergence Relabltyanalyss:aparametrcstudy Bucklngmodel(correlatednongaussanvarables) Probablstcanalyss:analytcalapproach Relabltyanalyss:analytcalapproach Relablty analyss: llustraton of numercal resoluton (AFOSMC) Probablstcanalyss:analytcalapproach Relabltyanalyss:analytcalapproach Relablty analyss: llustraton of numercal resoluton (AFOSMC) Vbratonmodel(nonlnearfalurefuncton) Probablstcanalyss:analytcalapproach Relablty analyss: llustraton of numercal resoluton (FOGSM) Relablty analyss: llustraton of numercal resoluton (FOGAM) A Computer Programs 76 A.1 AFOSMC A.1.1 CalltoAFOSMC A.1.2 AFOSMCsubroutne A.2 FOGSM A.2.1 CalltoFOGSM

3 CONTENTS Denoel Vncent, An ntroducton to Relablty Analyss A.2.2 FOGSMsubroutne A.2.3 FOGSM_InvTransfsubroutne A.3 MonteCarlo(nonGaussan) A.3.1 CalltoMCS_NG A.3.2 MCS_NGsubroutne

4 Chapter 1 Introducton Snce the early tmes of cvl engneerng the desgn of structures has been perfomed n a determnstc way,.e. under the assumpton of gven loads actng on structures wth gven propertes, whch results n unque dsplacements and nternal forces. Durng the whole lfe of the structure t should however be accepted that the loadng s not unque and that the materal propertes are not accurately determned n advance. For these reasons a certan varablty n the loadng as well as n structural propertes could have to be taken nto account. Ths results n a probablstc desgn. Usually the desgn of a structure conssts n two successve steps: frst the analyss and then the verfcatons: Desgn = Analyss + Verfcatons In a determnstc desgn these two steps come naturally the one after the other: the purpose of the analyss s to compute the structural dsplacements and nternal forces (or stresses); then the verfcatons am at checkng some desgn crtera as resstance or dsplacement exeedance. These two steps can easly be separated because the data to be conveyed from the analyss to the verfcatons are smple determnstc values: unque dsplacements and stresses. In a stochastc desgn, the separaton between the two steps s not necessarly obvous. Indeed snce the nput data of the problem s not determnstc anymore, the purpose of a probablstc analyss s to provde a probablstc descrpton of the structral response nstead of a unque determnstc value. Ths s llustrated at Fgure 1.1: the loads actng on the structure, the structural behavour and eventually the geometry are all represented by gven probablty densty functons (Fg. 1.1-a). The results of the analyss are the probablty densty functons of the dsplacements and nternal stresses at each pont of the structure (Fg. 1.1-b). These results can be acheved: by analytcal developements and hand calculatons n some smple cases, by a Monte Carlo Smulaton (MCS) technque, by means of fuzzy arthmetcs, by means of a stochastc fnte element approach. The MCS technque s the most tme-consumng method but leads, wthout any hypothess, to the most complete results. It s able to provde the full probablty densty functon of the response and hence any subsequent result. Once the determnstc theores are well understood ts applcaton s straghtforward. Concernng the method based on fuzzy arthmetcs t smply conssts n common operatons (+, -, *, /) but appled on fuzzy numbers. Ths methods allows thus computng the fuzzness of the response (.e. ts varablty whch s equvalent to the probablty densty functon). 3

5 CHAPTER 1. Denoel Vncent, An ntroducton to Relablty Analyss INTRODUCTION Fgure 1.1: Probablstc Analyss (a) The random loadng and random materal propertes are specfed by ther probablty densty fucntons (b) The result of the analyss s the probablty densty functons of the dsplacements 4

6 CHAPTER 1. Denoel Vncent, An ntroducton to Relablty Analyss INTRODUCTION Because of the evdent complexty of some problems, the analytcal approach can t be appled n any case; because of ther evdent tme demand and manly because of the huge quantty of nformaton comng out from the analyss, the MCS technque and the fuzzy arthmetcs are rather used for some localzed checkngs. For these reasons, n cvl engneerng applcatons, the stochastc analyss and the stochastc verfcatons are generally proceeded at once n a so-called relablty analyss 1. The use of ths method s also explaned by the fact that, at the desgn stage, the desgner s seldomly nterested n the whole pdf s of structural dsplacements but well n the upper tals of them. Indeed the upper tals of the pdf s only wll be used for the desgn. The dea s thus to focus on ths area only and not on the whole pdf s: the analyss conssts n evaluatng the relablty of the structure,.e. the probablty that, gven some probablstc loadng and structural parameters, the load appled on the structure overcomes ts resstance. Ths defnton shows that ths procedure nvolve both analyss and verfcatons at the same tme. Compared to a Monte Carlo smulaton, ts applcaton s very fast. As llustrated below t can ndeed be seen as computng one sngle pont of the whole pdf whereas,n ts most basc formulaton, the MCS technque establshes t all. A relablty analyss conssts n both the analyss and verfcaton of the structure. Exactly as for the verfcaton stage of a determnstc desgn, a set of checkng condtons have to be provded. In order to smplfy the problem, n ths document, these condtons are consdered one by one and the structure s sad to be safe f all the condtons are fulflled. For the sake of smplcty one condton only wll be consdered n ths report. Altough n a determnstc approach the checkng condton can be proved to be fulflled or not.e. to the queston "Is the resstance larger than the load?", the answer s yes or no, n a stochastc procedure, ths condton can t be assessed otherwse than wth a probablstc measure: the probablty of falure. The answer s of ths knd: "yes the resstance s larger than the load wth a probablty equal to 95%". In the very smple context of two varables, resstance R and loadng S, the falure condton s: Z = R S (1.1) The probablty of falure s expressed by: ZZ p f = prob(z <0) = p RS (R, S) drds (1.2) (R,S) Z<0 and ts computaton s the major am of any relablty analyss. 1 Because t combnes both analyss and verfcatons t should be rather called a "relablty-based desgn" 5

7 Chapter 2 Relablty analyss In a determnstc desgn, and n the context of a two-varable problem, resstance R and loadng S, the falure condton s assessed by the relaton: S>R (2.1) If ths condton s not fulflled, the falure s sad to occur. Wthn the context of such a determnstc approach, ths nequalty can also be wrtten wth several equvalent formulatons. For nstance: R S<0 or R S 1 < 0 (2.2) are strctly equvalent to Eq. (2.1). Although n a determnstc approach the checkng condtoncanbeprovedtobefulflled or not.e. to the queston "Is the resstance larger than the load?", the answer s yes or no, n a stochastc procedure, ths condton can t be assessed otherwse than wth a probablstc measure: the probablty of falure. The answer s then of ths knd: "yes the resstance s larger than the load wth a probablty equal to 95%". In a relablty analyss, a falure condton has to be defned too. It s drectly derved from the determnstc relaton and the falure condton s defned by a new random varable G. For example the random varables correspondng to the determnstc relatons n (2.2) are: G 1 (R, S) =R S or G 2 (R, S) = R S 1 (2.3) In ths smple context of a two-varable problem the falure condton s: G (R, S) < 0 (2.4) and the probablty of falure s thus expressed by: ZZ p f = prob(g <0) = p RS (R, S) drds (2.5) (R,S) G<0 where p RS (R, S), the jont probablty densty functon between R and S, has to be ntegrated on the doman on whch the falure condton s fulflled (where the resstance s smaller than the loadng). Ths s llustrated at Fg In practcal applcatons the resstance of the structure s gven as a functon of several parameters dependng on the consdered problem (stffness, cross-secton area, bendng modulus, structural dmensons, etc.). Then n a more general context a falure condton s gven as a scalar mplct relaton of more than two parameters: G({x}) =G(x 1,x 2,..., x N ) (2.6) 6

8 Frst Order Second Moment (FOSM) Fgure 2.1: Illustraton of the relablty analyss on 2-D varable problem. the purpose of a relablty method conssts n estmatng the probablty of falure as a result of the ntegral: Z Z p f = prob (G({x}) 0) = p x (x 1,x 2,...,x N )dx 1 dx 2 dx N (2.7) {x} G({x}) 0 where p x (x 1,x 2,..., x N ) s the jont probablty densty functon between all the varables. In case of uncorrelated, gaussan varables and lnear falure functon the results of ths ntegral can be obtaned n close form. However n practcal applcatons, these three condtons are very seldom satsfed together. The next sectons wll present how approxmate methods have been developed n order to gve estmatons of ths ntegral. The Frst Order Relablty Methods (FORM) conssts n a large set of avalable methods and some of them are presented. The most basc one, the Frst Order Second Moment, s presented and ts lack of rgour s enlghtened. Then Advanced Frst Order Second Moment (AFOSM) are ntroduced as a result of the Hasofer-Lnd theory. At ths stage the smple theory vald for Gaussan Processes as well as the Equvalent Gaussan Method s presented. 2.1 Frst Order Second Moment (FOSM) Let us assume that the load S and resstance R are two uncorrelated gaussan varables. They are characterzed by ther mean values (µ S and µ R ) and standard devatons (σ S and σ R ).Let us consder agan the falure condton of Eq. (2.3): Z 1 (R, S) =R S or Z 2 (R, S) = R S 1 (2.8) where G has been re-wrtten Z because Z 1 s a gaussan varable. Indeed, the theory of probablty states that ths lnear combnaton of two gaussan varables s stll gaussan. Z 1 s then very easy to handle: ts mean and standard devaton are gven by: µ Z = µ R µ q S (2.9) σ Z = σ 2 R + σ2 S (2.10) The probablty of falure, defned as the probablty that Z 1 s negatve (Eq. (2.5)), s represented by the shaded area n Fg. 2.2 and can be computed by: µ µ µz µz p f = Φ =1 Φ (2.11) σ Z 7 σ Z

9 Frst Order Second Moment (FOSM) Fgure 2.2: Probablty densty functon of Z 1 = R S where Φ s the normal cumulatve dstrbuton functon. Ths relaton shows that t s very convenent to defne a relablty ndex as: β = µ Z = µ R µ pσ S σ 2 Z R + σ 2 S (2.12) so that the probablty of falure s expressed by: p f = Φ (β) =1 Φ (β) (2.13) Hstorcally the frst dea of the relablty analyss was to compute ths relablty ndex as the rato of the mean falure condton and ts standard devaton, and then to express the probablty of falure by Eq. (2.13). Ths development shows that ths way to estmate the probablty of falure s rgorous,.e. returns the exact probablty of falure, n case of uncorrelated gaussan varables and lnear falure functon. Because t was developed from the consderaton of a lnear falure functon (Frst Order) and Gaussan processes (represented by ther frst two statstcal moments), ths method s called the Frst-Order Second-Moment method (FOSM). Ths procedure for the computaton of the probablty of falure can be used for the second "equvalent" falure condton Z 2 = R S 1 < 0. In ths case, the mean value of the falure condton µ Z and ts standard devaton σ Z are not easy to compute s close form. They can however be estmated easly f the falure condton s lnearzed around the mean value: Z 2 (R, S) ' Z 2 (µ R,µ S )+(R µ R ) Z 2 R The expectaton of ths relaton, taken sde-by-sde, gves: º +(S µ R=µ S ) R S=µ S Z 2 S º (2.14) R=µ R S=µ S µ Z2 = Z 2 (µ R,µ S )= µ R µ S 1 (2.15) and the varance of Z 2 s expressed by: σ 2 Z 2 = E (R µ R ) Z º 2 +(S µ R R=µ S ) R S=µ S º º = σ 2 Z 2 R + σ 2 Z 2 R R=µ S R S R=µ R S=µ S S=µ S º Z 2 S R=µ R S=µ S 2 = σ2 R µ S µ Rσ 2 S µ 2 S (2.16) 8

10 Frst Order Second Moment (FOSM) Fgure 2.3: Example of lnear falure functon and thus lnear falure condton The relablty ndex, as defned n the FOSM method, s: β = µ Z 2 σ Z2 = µ R µs 1 r σ 2 R µs µ R σ2 S µ 2 S (2.17) The probablty of falure related to ths second falure condton s then expressed by p f = Φ (β) =1 Φ (β). Smply because the relablty ndces are dfferent (Eqs. (2.12) and (2.17)) the probablty of falure s dfferent than the one obtaned wth the frst falure condton. Ths result ndcates that the FOSM method has to be used wth an extreme care. Indeed t would be expected that several falure condton gvng the same result n a determnstc approach gve also the same result n a probablstc approach. Ths example shows that t s not the case wth the FOSM method. Ths s known as the lack of nvarance property and s llustrated n secton Defnton 1 A relablty analyss method presents the nvarance property f any transformaton of a lnear falure functon returns the same relablty ndex. It s mportant to notce that the nvarance property s related to a lnear falure condton and not necessarly to a lnear falure functon. Ths dfference seems to be subtle but s very mportant. Indeed a non lnear falure functon, e.g. Z 2 (R, S) =R/S 1, can be assocated to a lnear falure condton (Z 2 =0 R S =0whch s lnear).ths s llustrated at Fgs 2.3 to 2.5. The frsttwoarelnearandnon lnear falure functons but leadng to the same lnear falure condton: Z =0s the same straght lne on both graphs. Typcally non lnear falure functon related to lnear condtons (Fg. 2.4) can be obtaned by a transformaton of a lnear falure functon. The nvarance property concerns these functons. Fg. 2.5 llustrates a non lnear functon wth a non lnear falure condton. In ths case the lmt condton Z =0s not lnear. The FOSM method s rgorous n case of lnear falure functon only (Fg. 2.3 only) and not for any lnear falure condton. Therefore t does not fulfl the nvarance property. The FOSM method conssts n replacng the exact falure functon by a lnearzed relaton, ths lnearzaton beng done at the mean pont. In case of lnear falure functon, ths lnearzaton returns the same result no matter the pont around whch the lnearzaton s performed. But Fg. 2.4 shows clearly that f the lnearzaton s performed on a pont whch does not lye on Z =0, the resultng hyperplane s dfferent (because level lnes are not parallel). Ths s the exact reason for whch the FOSM presents the lack of nvarance. Despte ths major drawback the FOSM method has been wdely used for many years. Indeed f the random varables are Gaussan and f the falure functon s lnear ths method returns the 9

11 Frst Order Second Moment (FOSM) Fgure 2.4: Example of non lnear falure functon but lnear falure condton Fgure 2.5: Example of non lnear falure functon and non lnear falure condton 10

12 Advanced Frst Order Second Moment (AFOSM) exact probablty of falure. If the probablty dstrbuton are slghtly dfferent than the gaussan dstrbuton or f the falure condton s slghtly non lnear, ths method can however be used to determne estmatons of the exact probablty of falure. The mathematcal defnton of the FOSM method can be generalzed n case of several random varables. Defnton 2. FOSM: Let us suppose a non lnear relaton of random varables: G({x}) =G(x 1,x 2,..., x N ) (2.18) descrbng the falure condton G({x}) =0. The relablty ndex related to ths falure condton s defned as the rato of the mean and standard devaton of the falure functon: β = µ G (2.19) σ G where the mean and standard devaton are estmated by lnearzng the falure condton around the mean varable {µ x } : whch gves: G({x}) ' G ({µ x })+({x} {µ x }) T =1 j=1 G º {x} {x}={µ x } (2.20) µ G = G({µ x }) (2.21) σ 2 G = NX NX º º G G cov XX x {x}={µ x } x j j {x}={µ x } (2.22) where cov X X j represents the covarance between varables X and X j. The subsequent probablty of falure s rgorous n case of gaussan varable and lnear falure functon only. 2.2 AdvancedFrstOrderSecondMoment(AFOSM) Presentaton of the method Because of ts lack of nvarance the Frst Order Second Moment method has been upgraded to an advanced method. The dea s then to develop a relablty method able to return the exact probablty of falure n case of any lnear falure condton and not only lnear functon (e.g as n Fg. 2.4). The major reproach that can be made to the prevous development s that t consders from the begnnng the lnearty of the falure functon and not only the lnearty of the falure condton. Instead of focusng on the probablty dstrbuton of G({x}) and manly ts mean and standard devaton, other methods rather target the estmaton of the probablty of falure as a drect result of the multple ntegral (Eq. (2.5)). As t s presented n the followng ths lead to relablty methods havng the nvarance property. They are known as Advanced Frst Order Second Moment (AFOSM) relablty analyses. Smlarly to the developments of the prevous secton, the most smple AFOSM s presented n the context of uncorrelated Gaussan varables and lnear falure condton.frst t s convenent to transform the physcal varables nto zero-mean and unt standard devaton varables (Hasofer- Lnd transformaton). New reduced varables are defned by: bx = x µ x (2.23) σ x whch have now the nterestng property to lead to an axsymmetrc jont probablty densty functon (because the ntal varables are supposed to be uncorrelated). The falure condton s supposed to be lnear,.e. t can be wrtten n ths form: G({x}) =0 a 1 x 1 + a 2 x a N x N = a 0 {a} T {x} = a 0 (2.24) 11

13 Advanced Frst Order Second Moment (AFOSM) Ths falure condton can also be wrtten as a functon of the new varables as: bg({bx}) =G(x 1 (bx 1 ),x 2 (bx 2 ),..., x N (bx N )) (2.25) Snce the orgnal falure condton was lnear the lnear transformaton keeps the lnearty of the lmts of the doman of ntegraton. Then b G({bx}) can be wrtten: bg({bx}) =0 ba 1 bx 1 + ba 2 bx ba N bx N = ba 0 {ba} T {bx} = ba 0 (2.26) Ths s schematcally llustrated on the 2-varable problem n Fg Because the varables are uncorrelated, the prncpal axes of the jont probablty densty functon between R and S are parallel to axes R and S (Fg. 2.6-(a)). The shaded area represents the doman of ntegraton,.e. the zone n whch G<0, or agan the doman n whch the resstance R s smaller than the loadng S. Wth the new reduced varables (Fg; 2.6-(b)), the jont probablty densty functon s represented by concentrc crcular level curves and the falure condton Z b =0s stll a lnear relaton of R b and S. b Furthermore n the context of uncorrelated varables the jont probablty densty functon can be factorzed n ts margnal probablty densty functons as: Z Z p f = p x1 (x 1 )p x2 (x 2 )...p xn (x N )dx 1 dx 2 dx N (2.27) {x} G({x}) 0 or as a functon of the reduced varables: Z Z p f = p bx1 (bx 1 )p bx2 (bx 2 )...p bxn (bx N )dbx 1 dbx 2 dbx N (2.28) {bx} bg({bx}) 0 where p bx (bx )=p x (x (bx )), [1,N] are the margnal probablty densty functons of the reduced varables,.e. the normal dstrbutons because the dstrbuton of the physcal varables s gaussan. By notng the normal probablty densty functon as p z (x), Eq. (2.28) can thus also be wrtten: Z Z p f = p z (bx 1 )p z (bx 2 )...p z (bx N )dbx 1 dbx 2 dbx N (2.29) {bx} b G({bx}) 0 The factorzaton of the jont probablty densty functons reduces the complexty of the defnte ntegral to the shape of ts doman of ntegraton only. Ths shape s however not really complex because ts lmt, represented by the falure condton, s a hyperplane (a lne n Fg. 2.6-(b)). In order to make t smpler agan, let us defne another transformaton: {ex} =[R] {bx} (2.30) where [R] s the adequate rotaton matrx transformng the doman of ntegraton nto a doman wth axes parallel to the reduced drectons. It can be understood ntutvely that the probablty of falure can be expressed as: p f = Z + p z (ex 1 )dex 1 Z + Z + p z (ex N 1 )dex N 1 p z (ex N )dex N (2.31) where the doman of ntegraton s runnng from to + for each varable except for the last one whch has to cover the nterval [β,+ ]. Because of Kolmogorov s frst axom and because the margnal probablty densty functons are normal, the probablty of falure s thus fnally reduced to: Z + p f = p z (ex N )dex N =1 Φ (β) =Φ ( β) (2.32) β β 12

14 Advanced Frst Order Second Moment (AFOSM) Fgure 2.6: Successve transformatons: (a) Intal physcal varables - (b) Reduced varables wth zero mean and unt varance - (c) Rotated reduced varables 13

15 Advanced Frst Order Second Moment (AFOSM) where Φ s the normal cumulatve densty functon and β s the relablty ndex defned as the shortest dstance, n the reduced space (Fg. 2.6-(b)), between the orgn and the falure condton. On the smple two-varable problem wth gaussan varables and lnear falure condton, ths geometrc defnton of the relablty ndex s equvalent to what was gven n the prevous secton (FOSM). It s however dfferent n case of a non lnear falure condton as llustrated n secton Both the FOSM and AFOSM gve the same relablty ndces and hence probablty of falure n case of gaussan processes and lnear falure functons. If the falure functon s non lnear but the falure condton s lnear the AFOSM return also the exact probablty of falure. Ths can be understood ntutvely snce the AFOSM method s based on a geometrc approach and the doman of ntegraton s anyway lmted by a hyperplane n case of lnear condton (no matter the lnearty or not of the functon). Furthermore the lnearty of the falure functon s not sought at all n the AFOSM whereas t was the FOSM s based on ths hypothess. In case of non lnear condtons, the AFOSM method gves however estmatons only of the exact probablty of falure. Because of the geometrc defnton of the relablty ndex, the AFOSM method can be appled but t should be kept n mnd that the resultng error n the probablty of falure ncreases as the non lnearty of the condton ncreases. Defnton 3. AFOSM: Let us suppose a non lnear relaton of uncorrelated Gaussan varables: G({x}) =G(x 1,x 2,..., x N ) (2.33) descrbng the falure condton. A set of reduced (zero-mean unt-varance) varables s defned by: bx = x µ x (2.34) σ x and the falure condton, expressed wth these new varables s: bg({bx}) =G(x 1 (bx 1 ),x 2 (bx 2 ),..., x N (bx N )) = 0 (2.35) The relablty ndex related to ths falure condton s defned as the shortest dstance n the reduced space between the orgn and ths hypersurface. Theorem 4 To fnd the mnmum of a scalar functon F ({bx}) under the condton G ({bx}) =0 s a usual mathematcal problem. It s usually solved by ntroducng Lagrange multplers. Indeed for an extremum of F to exst on G the gradent of F must lne up wth the gradent of G. Ones then the multple of the other OF = λog (2.36) where λ s called the Lagrange multpler. If these two vectors are equal then each of ther components are also equal: F + λ G =0 for =1,...N (2.37) x x Together wth G ({bx}) =0,theseN relatons form a set of N +1 equatons wth N +1 unknowns. The optmum pont s obtaned by the resoluton of ths set of non lnear equatons. The AFOSM requres the computaton of the shortest dstance between a gven hypersurface and the orgn. Ths can be performed by ntroducng a Lagrange multpler. The dstance between any pont and the orgn s expressed by: q β ({bx}) = {bx} T {bx} (2.38) The applcaton of Lagrange s theory to the squared dstance lead to ths set of equatons: ( bx ³{bx} T {bx} + λ G b bx =0 bg (bx 1, bx 2,..., bx N )=0 (2.39) 14

16 Advanced Frst Order Second Moment (AFOSM) whch gves, after smplfcaton: ( b G bx =0 bx + λ 2 bg (bx 1, bx 2,..., bx N )=0 (2.40) The resoluton of ths set of equatons gves the desgn pont (or most probable falure pont) {bx }. The correspondng relablty ndex can subsequently be estmated thanks to Eq. (2.38) Lnk between the FOSM and AFOSM methods The FOSM method s based on the lnearzaton of the falure functon at the mean varable pont ({x} = {µ x }). Because ths pont does not necessarly le on the falure lmt, ths relablty method does not have the nvarant property. In order to bypass ths problem the lnearzaton of the functon could be done at the desgn pont ({x} = {bx }) whch les n any case on the falure lmt. The falure functon s thus approached by: º G({x}) ' ({x} {x }) T G (2.41) {x} {x } n whch G ({bx })=0has been taken nto account. It s mportant to notce that the desgn pont {bx } s a pror unknown. Followng the FOSM s procedure, the relablty ndex s expressed as the rato of the mean value of the falure condton µ G and ts standard devaton σ G. Thanks to the new lnearzaton they are expressed by: º µ G = E [G({x})] = ({µ x } {x }) T G (2.42) {x} {x } and σ 2 G = E h(g({x}) µ G ) 2 Ã º! 2 = E ({x} {µ x }) T G {x} {x } = = NX =1 j=1 NX =1 j=1 NX h x E µ x NX cov x x j β = µ G σ G = º G x {x } ³ º G x j µ xj x {x } º G x j {x } º G x j {x } where cov XX j represents the covarance between varables X and X j. In order to compare more easly wth the development related to the AFOSM method the same development can be wrtten n the reduced space ({µ x } 0, cov x x j δ j, x bx). The resultng relablty ndex s then expressed by: P N =1 bx bg bx k{bx } s µ P N bg =1 bx k{bx } 2 (2.43) In the AFOSM method the relablty ndex s defned as the shortest dstance between the orgn (n the reduced space) and the lmt hyperplan. It has been shown that the resultng expresson of the relablty ndex s (Eq. (2.38)): v ux β = t N bx 2 (2.44) =1 15

17 Advanced Frst Order Second Moment (AFOSM) After some trval transformatons and usng Eq. (2.40), ths relaton can also be wrtten: v ux β = t N P 2 N bx 2 λ =1 = bx 2 =1 =1 bx q PN =1 = P N 2 λ q PN =1 bx 2 2 λ bx = 2 2 λ bx P N =1 bx b G bx k{bx } s µ P N bg =1 bx k{bx } 2 (2.45) whch results n the same relaton as Eq. (2.43). Ths ndcates clearly that the geometrc defnton of the relablty ndex (wthout requrng any lnearzaton of the falure functon!) s strctly equvalent to the lnearzaton of the falure functon around the desgn pont. Ths demonstraton establshes a drect and clear lnk between the FOSM method and the AFOSM method. A second defnton can thus be gven for the AFOSM method. Defnton 5. AFOSM: Let us suppose a non lnear relaton of random varables: G({x}) =G(x 1,x 2,..., x N ) (2.46) descrbng the falure condton G({x}) =0. The relablty ndex related to ths falure condton s defned as the rato of the mean and standard devaton of the falure functon: β = µ G (2.47) σ G where the mean and standard devaton are estmated by lnearzng the falure condton around the desgn pont {x }: whch gves: G({x}) ' ({x} {x }) T º µ G = ({µ x } {x }) T G {x} NX NX º σ 2 G G = cov x x j x =1 j=1 G º {x} {x}={x } {x}={x } {x}={x } G x j º {x}={x } (2.48) (2.49) (2.50) where cov X X j represents the covarance between varables X and X j. The subsequent probablty of falure s rgorous n case of gaussan varable and lnear falure condton only. Snce the lnearzaton s performed around an a pror unknown pont (the desgn pont), an teratve resoluton scheme has to b adopted Detals of the resoluton The prevous developments related to the AFOSM show that ths relablty method can be consdered n two equvalent (but clearly dstnct) ways: the geometrc defnton of the relablty ndex (Def. 3) leads to a set of non lnear equatons. Ths system has to be solved (2.40) wth any numercal procedure. Usually a second-order teratve scheme, lke Newton-Raphson method, s used. the relablty ndex can be obtaned as the rato of the mean to the standard devaton of the falure functon, after havng lnearzed t around the desgn pont (Def. 5). Snce the desgn pont s a pror unknown, ths procedure requres also an teratve technque for the computaton of the relablty ndex. 16

18 Advanced Frst Order Second Moment (AFOSM) The equvalence between these two methods s seldom presented and many author focus just on one or the other wthout any justfcaton. In ths report, the equvalence has been demonstrated (Secton 2.2.2) and t s clear that both ways to estmate the relablty ndex wll gve the same result 1. For ths reason, n the followng, the second way to consder the relablty ndex wll be adopted. In hs lecture note on relablty analyss ([5]), Kusama adopts ths alternatve and apples a convenent teratve resoluton procedure. The successve steps of ths procedure can be summarzed n an algorthmc manner. Algorthm 6 AFOSM n the physcal space, for uncorrelated gaussan varables Step 1: Gve an ntal guess 2 of the desgn pont x (0) and start the teratons at the next step (wth k =0). Step 2: There s no reason for ths pont to le on the falure condton. So, compute the falure condton at ths pont: ³n (k)o G (k) = G x (2.51) Step 3: Compute the gradent of the falure functon at ths pont: n (k) = G({x}) º x {x}={x (k) } (2.52) Step 4: Compute a new orentaton for the desgn pont: α (k) = v ux t N =1 n (k) σ x ³ n (k) σ x 2 (2.53) Step 5: Compute the estmated mean and standard devaton of the falure condton: µ G = G (k) + σ G = NX =1 NX ³ n (k) µ x x (k) =1 Step 6: Compute a new estmaton of the relablty ndex: (2.54) α (k) n (k) σ x (2.55) β (k) = µ G (2.56) σ G Step 7: Compute a better estmaton of the desgn pont: x (k+1) = µ x α (k) β (k) σ x (2.57) ncrement k by 1 andloopfromstep2tostep7untltheconvergencesreached. 1 The major dfference between both ways to consder the relablty ndex concerns the convergence of the teratve schemes. For some badly condtoned systems (.e. some complex falure condtons), one method could be much faster than the other, or even converge correctly whereas the other dverges dramatcally. 2 The ntal guess of the desgn pont could smply be the mean value (the frst teraton s then the exact applcaton of the FOSM method). In Kusama s procedure the ntal guess of the desgn pont s gven by ntal guesses on β (0) and α (0)ª where α represent the orentaton of the desgn pont wth respect to the mean values. As a frst guess, he recommends to choose β (0) = 3 and α (0) = 1 whch seems to be approprate n some N crcumstances only. The ntal desgn pont s then computed by: x (0) = µ x α (0) β (0) σ x 17

19 Advanced Frst Order Second Moment (AFOSM) Fgure 2.7: Illustraton of the dfference between: (a) the physcal space (Algorthm 6) and (b) the reduced space.(algorthm 7). The developments of the next sectons wll ndcate that the use of reduced varables s not really necessary but clearly convenent n case of correlated varables. For ths reason, ths modfcaton s already adopted at ths stage. Algorthm 6 s slghtly modfed n order to perform the relablty analyss n the reduced space (zero-mean and unt-varance varables). Algorthm 7 AFOSM n the reduced space, for uncorrelated gaussan varables Step 0.1: Defne reduced varables: bx = x µ x = µ σ + σ bx (2.58) Step 0.2: Wrte the falure condton wth the reduced varables: bg ({bx}) =G ({x ({bx})}) (2.59) Step 1: Gve an ntal guess of the desgn pont bx (0) and start the teratons at the next step (wth k =0). Step 2: There s no reason for ths pont to le on the reduced falure condton. So, compute the reduced falure condton at ths pont: bg (k) = G b ³n (k)o bx (2.60) Step 3: Compute the gradent of the reduced falure functon at ths pont: n (k) = G b % ({bx}) bx {bx}={bx (k) } Step 4: Compute a new orentaton for the desgn pont: α (k) = n(k) n (k) n (k) = v ux t N ³ Step 5: Compute the estmated mean and standard devaton of the falure condton: µ G = b G (k) σ G = NX =1 =1 NX =1 n (k) 2 (2.61) (2.62) n (k) x (k) (2.63) α (k) n (k) (2.64) 18

20 Advanced Frst Order Second Moment for Correlated varables (AFOSMC) Step 6: Compute a new estmaton of the relablty ndex: β (k) = µ G (2.65) σ G Step 7: Compute a better estmaton of the desgn pont: bx (k+1) = α (k) β (k) (2.66) ncrement k by 1 andloopfromstep2tostep7untltheconvergencesreached. The selecton of the frst guess for Step 1 of ths algorthm s subject to the same remarks as n Algorthm 6. However t should be added that f the frst guess s taken as the mean physcal varables (eventually reduced), Algorthms 6 and 7 correspond to the smple FOSM after the frst step. Ths algorthm and ths remark are presented by an llustratve example n secton (p. 34). The applcaton of ths algorthm s rgorous n case of lnear falure condton only. In the opposte case, the method can also be appled but t should be kept n mnd that t gves approxmate results only. 2.3 Advanced Frst Order Second Moment for Correlated varables (AFOSMC) Under the three condtons consdered n secton 2.2 (uncorrelated, gaussan varables and lnear falure condton) the probablty of falure s drectly related to the relablty ndex β through Eq. (2.32). If any one of these condtons s not fulflled the relablty ndex can be computed (thanks to ts geometrcal defnton) but can t be lnked drectly to the probablty of falure. For ths reason the relablty of a structure s often quantfed by ts relablty ndex rather than by ts probablty of falure. However the am of many relablty analyss methods s to relate as accurately as possble the relablty ndex to the actual probablty of falure of the structure. Ths means also that f any one of the three condtons consdered n the context of Algorthm 7 s not fulfled, ths procedure can however be used but leads defntely to erroneous estmatons of the actual relablty ndex and probablty of falure. More approprate resoluton technques have to be consdered n ths case. Ths secton ams at presentng the modfcatons to be brought to the AFOSM method (Algorthm 7) n order to use t n the context of correlated varables. Ths results n the AFOSMC method. When the physcal varables (x ) nvolved n the problem are not uncorrelated, the prncpal axes of the jont probablty densty functon are no longer parallel to the axs (Fg.2.8-(a)). The smple reducton proposed by Eq. (2.23) does not transform the jont probablty densty functon nto an axsymmetrc functon. To ths purpose, another transformaton has to be ntroduced: {bx} =[A]({x} {µ x }) (2.67) where {µ x } = E [{x}] s the vector of mean physcal varables and [A] s the transformaton matrx whch wll produce uncorrelated reduced varables {bx}. The mathematcal expectaton of ths relaton shows that ths transformaton provdes zero-mean reduced varables. Furthermore ther covarance matrx s expressed by: [V bx ]=E h {bx}{bx} T =[A] E h ({x} {µ x }) ³ {x} T {µ x } T [A] T =[A][V x ][A] T (2.68) Snce the am of the reducton s to provde zero-mean and unt-varance varables, the matrx [A] wll be chosen such as to provde: [V bx ]=[A][V x ][A] T =[I] (2.69) 19

21 Advanced Frst Order Second Moment for Correlated varables (AFOSMC) Fgure 2.8: where [I] s the dentty matrx. Hence [A] can smply be estmated as the egen vectors of the covarance matrx of the physcal varables. The normalzaton of the egen vectors s of course performed n such a way that Eq. (2.69) s satsfed. In ths secton t s stll supposed that the falure condton s a lnear functon of the physcal parameters (Eq. 2.24). Snce the transformaton used to produce uncorrelated reduced varables s lnear, the falure condton expressed wth the new varables remans agan lnear: G({x}) =0 {a} T {x} = a 0 bg({bx}) ³ =0 {a} T {µ x } +[A] 1 {bx} = a 0 {a} T [A] 1 {bx} = a 0 {a} T {µ {z } x (2.70) {z } {ba} T ba 0 The reduced varables are now uncorrelated, ther jont probablty densty functon s axsymmetrc and can be factorzed. Then the same developments as presented n the prevous secton are stll vald. In partcular, Eq. (2.29) stll holds, the relablty ndex wth the same defnton as prevously can be determned and t s agan lnked rgorously to the probablty of falure. In vew of these developments, the former algorthm of the AFOSM method can be slghtly transformed to sut the context of correlated varables. Algorthm 8 AFOSMC, for gaussan varables Step 0.1: Defne reduced varables: {bx} =[A]({x} {µ x }) {x} = {µ x } +[A] 1 {bx} (2.71) where [A] s such that [A][V x ][A] T varables {x}. =[I] and [V x ] s the covarance matrx of the physcal 20

22 Second-Order Relablty Methods (SORM) Step 0.2: Wrte the falure condton wth the reduced varables: ³ bg ({bx}) =G ({x ({bx})}) =G {µ x } +[A] 1 {bx} (2.72) Step 1: Gve an ntal guess of the desgn pont bx (0) and start the teratons at the next step (wth k =0). Step 2: There s no reason for ths pont to le on the reduced falure condton. So, compute the reduced falure condton at ths pont: bg (k) = G b ³n (k)o bx (2.73) Step 3: Compute the gradent of the reduced falure functon at ths pont: n (k) = G b % ({bx}) bx {bx}={bx (k) } (2.74) Step 4: Compute a new orentaton for the desgn pont: α (k) = n(k) n(k) = n (k) v ux t N ³ =1 n (k) 2 (2.75) Step 5: Compute the estmated mean and standard devaton of the falure condton: µ G = b G (k) σ G = NX =1 NX =1 Step 6: Compute a new estmaton of the relablty ndex: n (k) x (k) (2.76) α (k) n (k) (2.77) β (k) = µ G (2.78) σ G Step 7: Compute a better estmaton of the desgn pont: bx (k+1) = α (k) β (k) (2.79) ncrement k by 1 andloopfromstep2tostep7untltheconvergencesreached. The applcaton of ths algorthm s rgorous n case of lnear falure condton only. In the opposte case, the method can also be appled but t should be kept n mnd that t gves approxmate results only. If the physcal varables are uncorrelated, ths Algorthm degenerates to Alg. 7 and the reducton matrx [A] s a dagonal matrx composed of the nverses of the standard devatons of the random varables. Ths Algorthm s llustrated through worked examples n secton Second-Order Relablty Methods (SORM) When the falure condton exhbts strong non lneartes, the applcaton of FORM methods can lead to sgnfcant erroneous results. Indeed as ndcated n ts name, a FOR method consders the frst order representaton of the lmt state only,.e. replaces t by a hyperplane. Ths s llustrated 21

23 Second-Order Relablty Methods (SORM) Fgure 2.9: Illustraton of a non lnear falure functon and the correspondng lnearzed functon (at the desgn pont) 22

24 Frst-Order Gaussan Second Moment Method (FOGSM) n Fg Because of the defnton of the relablty ndex n a geometrc way, both falure condtons have the same relablty ndex. The probablty of falure s rgorously related to the relablty ndex when the falure condton s lnear and when the physcal varables are normally dstrbuted. For ths reason, the securty of usual structures aganst falure s often assessed by means of the relablty ndex only. Fg. 2.9 shows also that computng the probablty of falure assocated to a non lnear condton conssts n replacng t by the tangent hyperplane at the desgn pont (n ths example both falure condton Z 1 and Z 2 have the same relablty ndex). The resultng probablty of falure s thus underestmated f the curvature of the actual falure condton faces the orgn (as n Fg. 2.9); on the opposte, a curvature towards nfnty wll results n a overestmaton of the exact probablty of falure. The evaluaton of the probablty of falure n case of some smple non lnear falure functons can be found n the lterature but the cases n whch ths ntegral can be computed are very seldom. Anyway the explct computaton of the probablty of falure for any quadratc falure functon s not magnable! For ths reason, some approxmate methods have been developed. The most usual s due to Bretung ([1], [2]) whch gves a smple approxmate expressons of the probablty of falure, as a functon of the relablty ndex (obtaned wth a FORM method): n 1 Y 1 p f ' Φ ( β) p (1 βκ ) =1 n 1 Y p f ' Φ ( β) =1 1 r ³ 1 φ(β) Φ( β) κ (2.80) (2.81) The approxmate results are obtaned by asymptotcal developments of the ntegrands. The probablty of falure s also expressed as a functon of the extrnsc curvatures of the falure functon κ at the desgn pont. It can be checked that, f the curvatures are all equal to 0 (lnear falure condton), ths result degenerates nto Φ ( β) whch s well correct f the falure condton s lnear (FORM). The second relaton seems to provde better results because t smoothens the sngularty at βκ =1. In ths second expresson φ (β) denotes the standard normal densty functon. The development of such a method goes beyond the scope of ths techncal report and s therefore not llustrated n the followng examples. The error assocated to the use of a FORM n case of non lnear falure functon s however llustrated by comparson wth a Monte Carlo Smulaton (Sectons 3.2.6, and 3.3.3). 2.5 Frst-Order Gaussan Second Moment Method (FOGSM) The Gaussan dstrbuton s often used because of ts convenence. However not any physcal characterstc could be modelled wth a Gaussan varable. The most arguable pont s that a Gaussan varable can exhbt negatve (even, but wth a very small probablty). For ths reason, other random dstrbutons have to be consdered for the varables nvolved n a structural desgn. Fg 2.10-(a) llustrates the same falure condton (Z R S) as prevously consdered, but n the context of non gaussan varables. In ths case the level curves of the jont probablty densty functon are not ellptc anymore. An example of the most general case s gven n Fg (a). The prevous developments related to the FOSM and AFOSM methods ndcated that the probablty of falure could be computed easly, provded the jont probablty densty functon can be factorzed, allowng then a reducton of the order of ntegraton to 1 (and the defnton of the relablty ndex). Snce a huge quantty of developments has been performed wth ths usual defnton of the relablty ndex, a common way to handle non gaussan varables s to transform the problem nto the same form as prevously. An adequate transformaton (detals gven hereafter) s computed n such a way to transform the non gaussan varables nto gaussan 23

25 Frst-Order Gaussan Second Moment Method (FOGSM) Fgure 2.10: ones. Typcally ths reducton s a non lnear relaton (because any lnear functon of a gaussan process keeps ts lnearty). The falure condton s transformed accordngly (Fg (b)). Even f the orgnal falure condton was lnear (n the physcal space) the reduced falure condton exhbts a non lnear shape because of ths reducton. Then when the consdered random varables are non Gaussan, the reduced falure functon s non lnear. Adequate resoluton technques (e.g. SORM) should thus be used. However, provded the non lnearty remans slght, frst order methods (lke AFOSM) can already gve reasonable estmatons of the probablty of falure. Theorem 9 Transformaton of a sngle random varable. Let x and y be random varables wth ther respectve probablty densty functons p x (x) and p y (y). If y s expressed as a functon of x by a monotonc relaton y = y(x) whch leads to the unvoque reverse relaton x = x(y), both probablty densty functons are related by: p x (x) = p y (y) p y (y) = p x (x) dy(x) dx dx(y) dy (2.82) The proof of ths relaton s straghtforward when consderng the cumulatve densty functons. For example, the cumulatve densty functon of y s: F y (y 0 )=prob (y <y 0 )=prob (x <x(y 0 )) = F x (x (y 0 )) (2.83) 24

26 Frst-Order Gaussan Second Moment Method (FOGSM) and the correspondng probablty densty functon, obtaned by smple dervaton wth respect to y 0,s: p y (y) = df y (y) = df x (x) dx dy dx dy = p x (x) dx dy (2.84) Ths general relaton can be used n two nterestng context. Theorem 10 Tranformaton from a unform dstrbuton. Let us suppose that x s unformly dstrbuted on [0, 1] and that F y (y) s a gven cumulatve densty functon. The functon y(x) transformng x to y and such that the resultng cumulatve densty functon corresponds to F y (y) s: y (x) =Fy 1 (x) (2.85) If x s unformly dstrbuted on [0, 1], ts cumulatve densty functon s expressed by: 0 x 0 < 0 F x (x 0 )= x 0 x 0 [0, 1] 1 x 0 > 1 (2.86) Furthermore, from Eq. (2.83), t s possble to wrte: x (y 0 )=Fx 1 [F y (y 0 )] y (x 0 )=Fy 1 [F x (x 0 )] (2.87) whch demonstrates the theorem (F x (x 0 )=x 0 ). Ths property s commonly used for the generaton of non unform random varables (Monte Carlo smulaton technques). Indeed, today s computers are all equpped wth a random number generator. Ths generator s often lmted to the generaton of a unform random varable (between 0 and 1). Thanks to the prevous relaton any random varable wth a gven cumulatve densty functon (and hence probablty densty functon) can be generated. Theorem 11 Transformaton to a Gaussan dstrbuton. Let us suppose that the probablty dstrbuton of x s known. The transformaton y(x) that provdes a normal varable y s gven by: y(x) =Φ 1 [F x (x)] where Φ represents the gaussan cumulatve densty functon. The proof of ths theorem s a straghtforward applcaton of Eq. (2.87). Theorem 12 Transformaton of a multple random varables. Let {x} and {y} be sets random varables wth ther respectve jont probablty densty functons p x ({x}) and p y ({y}). If {y} s unvoquely expressed as a functon of {x} and reversely as well, both jont probablty densty functons are related by: p x ({x}) = p y ({y}) d {y({x})} d {x} = p y({y}). dy 1 dx N p y ({y}) = p x ({x}) d {x({y})} d {y} = p x({x}). dx 1 dy N dy 1 dx 1 dy 2 dx 1 dy N dx 1 dy 1 dx dx 1 dy 1 dx 2 dy 1 dy N dx N dx N dy 1 dx 1 dy dx N dy N (2.88) 25

27 Frst-Order Gaussan Second Moment Method (FOGSM) Fgure 2.11: Illustraton of transformaton functons. (a) From unform dstrbuton to Gaussan, (b) From the standard beta (2, 2) dstrbuton to Gaussan 26

28 Frst-Order Gaussan Second Moment Method (FOGSM) The demonstraton of ths relaton s smlar to the proof of Theorem 9. In case of correlated non gaussan varables, advanced tranformaton technques (Rosenblatt Transformaton, Nataf Transformaton) have to be appled. Ths goes beyond the scope of ths report. Then n the followng developments and llustratons t wll be supposed that the non gaussan random varables are not correlated. Ths allows reducng each of them separately wth Th. 11. Dependng on the orgnal probablty densty functon, the relaton y(x) whch wll provde a gaussan varable y has to be computed. Ths s llustrated at Fg for the unform dstrbuton and the standard beta dstrbuton. Each physcal varable leads then to the defnton of a correspondng reduced varable. The computaton of the relablty ndex s then performed as n the prevous case. These developments can be appled to formalze the method n an algorthmc manner. Algorthm 13 FOGSM n the reduced space, for uncorrelated gaussan varables Step 0.1: Defne reduced varables: bx = Φ 1 [F x (x )] x = Fx 1 [Φ (bx )] (2.89) In almost any case ths functon has to be computed n a numercal way. Step 1: Gve an ntal guess of the desgn pont bx (0) and start the teratons at the next step (wth k =0). Step 2: There s no reason for ths pont to le on the reduced falure condton. So, compute the reduced falure condton at ths pont: bg (k) = b G ³n bx (k)o = G ³n x (k) ³n bx (k)o o (2.90) Step 3: Compute the gradent of the reduced falure functon at ths pont: n (k) = G b % ({bx}) bx {bx}={bx (k) } (2.91) Step 4: Compute a new orentaton for the desgn pont: α (k) = n(k) n(k) = n (k) v ux t N ³ =1 n (k) 2 (2.92) Step 5: Compute the estmated mean and standard devaton of the falure condton: µ G = b G (k) σ G = NX =1 NX =1 Step 6: Compute a new estmaton of the relablty ndex: n (k) x (k) (2.93) α (k) n (k) (2.94) β (k) = µ G (2.95) σ G Step 7: Compute a better estmaton of the desgn pont: bx (k+1) = α (k) β (k) (2.96) ncrement k by 1 andloopfromstep2tostep7untltheconvergencesreached. Ths algorthm s appled n secton

29 Frst-Order Gaussan Approxmaton Method (FOGAM) 2.6 Frst-Order Gaussan Approxmaton Method (FOGAM) The developments of the prevous secton amed at presentng the rgorous procedure for the computaton of the relablty ndex n case of non gaussan processes. It was shown that the transformaton to the reduced space leads systematcally to a non lnear functon, even n case of lnear falure condton. For ths reason, the applcaton of a FORM method wll thus lead to approxmate results. Furthermore the frst step of ths method (transformaton to uncorrelated, zero-mean, untvarance varables) s not necessarly easy to perform wthout the approprate numercal tools. Snce the FOGSM does not provde the exact results, a smplfed verson of ths method s sometmes appled. In ths verson the complex transformaton s avoded. The key dea of ths method, the Frst-Order Gaussan Approxmaton Method, les n the replacement of the actual probablty densty functons by equvalent Gaussan probablty dstrbuton. There are several ways to perform ths equvalence but the most usual s presented n ths techncal report. Snce the most mportant regon concerns the falure doman, the equvalent gaussan varables are chosen such as to provde the same probablty densty functon and cumulatve densty functon at the desgn pont. For each varable two equatons allow thus the determnaton of the two necessary parameters (mean and varance) for the full charactersaton of the equvalent Gaussan process. Once equvalent Gaussan varable are establshed the problem s formatted as n the AFOSM method and the same resoluton procedure can be appled. Compared to the FOGSM, ths new method s a bt more tme-consumng. Indeed, the computaton of the equvalent gaussan varables depends on the locaton of the desgn pont and has thus to be performed at each teraton. Once the Algorthm 14 FOGAM n the reduced space, for uncorrelated gaussan varables Step 0.a: Gve an estmaton of the desgn pont x (0) and start the teratons at the next step (wth k =0). Step 0.b: Defne equvalent Gaussan varables (µ (0),σ (0) ) Step 0.1: Defne reduced varables: bx = x µ (0) σ (0) Step 0.2: Wrte the falure condton wth the reduced varables: x = µ (0) + σ (0) bx (2.97) bg ({bx}) =G ({x ({bx})}) (2.98) Step 1: Compute the correspondng reduced desgn pont bx (0) Step 2: There s no reason for ths pont to le on the reduced falure condton. So, compute the reduced falure condton at ths pont: bg (k) = G b ³n (k)o bx (2.99) Step 3: Compute the gradent of the reduced falure functon at ths pont: n (k) = G b % ({bx}) bx {bx}={bx (k) } (2.100) Step 4: Compute a new orentaton for the desgn pont: α (k) = n(k) n(k) = n (k) v ux t N ³ =1 n (k) 2 (2.101) 28