in Engineering Prof. Dr. Michael Havbro Faber ETH Zurich, Switzerland Swiss Federal Institute of Technology

Transcription

1 Risk and Saey in Engineering Pro. Dr. Michael Havbro Faber ETH Zurich, Swizerland

2 Conens o Today's Lecure Inroducion o ime varian reliabiliy analysis The Poisson process The ormal process Assessmen o he mean ou-crossing rae Hierarchical modelling in ime varian reliabiliy analysis Simpliicaions

3 I is imporan o emphasize ha probabiliies bilii are always somehow relaed o a ime measure Typically he ime measure can be - a paricular number o experimens - a ime inerval e.g. a year - a spaial characerisic e.g. lengh, area or volume In mos cases we can ormulae he reliabiliy problems in a way ha he ime measure does no explicily ener he probabilisic modelling o basic random variables In order o undersand when o do wha we will consider some basic aspecs o ime varian reliabiliy

4 As menioned earlier we can oen model reliabiliy problems such ha ime does no ener he probabilisic modelling o he basic random variables. This is e.g. he case when loads are ergodic in which case reliabiliy problems relaing o exreme load evens may be ormulaed using exreme value disribuions or he exreme load realisaions corresponding o a cerain ime inerval In such cases we may direcly use FORM analysis or he assessmen o he relevan probabiliies.

5 In order o be able o inroduce he basics o ime varian reliabiliy problems i is useul o inroduce wo special ypes o sochasic processes, namely he: - Poisson process - ormal process Used exensively o describe saisical characerisics o evens usually rare Used exensively o describe he ime varian behaviour o uncerain properies

6 The process denoing he number o poins in he ime inerval ;T[ is called a simple Poisson process i i saisies he ollowing condiions: The probabiliy o one even in he inerval ;D[ is asympoically proporional o D The probabiliy o more han one even in he inerval ;D[ is a uncion o a higher order erm o D or D Evens in disjoin inervals are sochasically independen The simple Poisson process may be described compleely by is densiy ν = lim Pone even in ; Δ[ Δ Δ For homogeneous Poisson processes, ν = consan

7 The probabiliy bili o n evens o a simple Poisson process in he inerval ;[ can be shown o be given by: P = n ν τ dτ n! n exp ν τ dτ From his, we can derive he probabiliy o no evens as: P = exp ν τ dτ and hen he probabiliy bili disribuion ib i or he ime ill he irs even as: FT = P = exp ν τ dτ

8 x Filered Poisson processes may be derived rom he simple Poisson process a a a We assume ha evens are generaed along he ime axis in accordance wih a simple Poisson process y y y y 4 Then we associae wih all such evens a response uncion ω,, Y k k I now we le he duraion approach zero, we ge he Poisson spike process x which is deined o be equal o or < k The ilered Poisson process is now esablished as: = k = ω, k,yk y y y y

9 x Here we consider he Poisson spike process wih muually independen random spikes Y i I e.g. ailure can be described as he even o a spike above he hreshold x y y y y I is recognised ha hese evens are also evens o a Poisson process wih inensiy ν = ν FY The probabiliy disribuion o he ime ill ailure hus becomes exponenially disribued P = exp ν τ F Y τ dτ

10 A random process is said o be ormal Gaussian i any se o random variables i, i =,,..n is joinly ormal disribued ed The mean number o ou-crossings o a random process above he hreshold can be derived as Rice s ormula The sample pahs mus be a leas one ime diereniable in respec o x, x Realisaion o ormal process ν = x x ϕ dx,, Firs order parial derivaive o x

11 For random processes in general we have where is he irs passage densiy uncion P T = ], T ] Δ = Δ τ Δτ = O Δτ ], T ] Δ P T = P P τ dτ x, Δ T Δ τ τ Δ P T = P P τ Δτ O Δτ ], T ] P T Δ Δ = P P ν d S τ ], T ] T

12 Time varian reliabily F l For ormal processes wo cases may be considered he saionar process and d x he saionary process and consan hreshold case exp exp dx x σ σ σ σ πσ ν = = he non-saionary process and/or p σ σ π he non-saionary process and/or non-consan hreshold case ω η ω η ω η ϕ η ϕ ω ν Φ = σ μ η = =, c σ σ ω

13 Approximaions o ime varian problems The irs passage ime is o ineres in reliabiliy applicaions which is exremely hard o assess P T = P P τ dτ T Whenever he reliabiliy problem can be ormulaed in erms o a condiional probabiliy o ailure given a cerain even he Poisson spike model can be used o approximae he irs passage probabiliy P = exp ν τ F τ dτ Y In cases where he probabiliy o ailure e.g. per annum may be calculaed, he simple Poisson process may be used o approximae he irs passage probabiliy FT = P = exp ν τ dτ

14 I is normally he case ha he ime varian includes no only random processes bu also random sequences and random variables I such cases he hierarchical model should be used Ou-crossing rae o he process F on-ergodic variables = E R exp E ν τ, R, Q dτ T Q Random sequences

15 In pracice i is useul o diereniae beween cases where he hreshold normalised Resisance and load b is slowly decreasing rapidly decreasing T Time Resisance and load Ou-crossing approach b End poin modelling T Time

16 problems may also be addressed approximaely by consideraion o sysems reliabiliy analysis F F F 3 F 4 F 5 Δ Δ Δ 3 Δ 4 Δ 5 P = P F F F F F 5 = P F F F3 F4 F5 3 4

17 A inal aspec o ime varian reliabiliy is he assessmen o he annual probabiliy o ailure or componens subjec o accumulaed deerioraion P Cumulaive ailure probabiliy In hese cases he reliabiliy analysis provides he ailure probabiliy as a uncion o he experienced service lie I ailures a consecuive imes may be assumed o be ully dependen he annual ailure probabiliies may be esablished by subracion P Annual ailure probabiliy