ASME 2013 IDETC/CIE 2013 Paper number: DETC A DESIGN ORIENTED RELIABILITY METHODOLOGY FOR FATIGUE LIFE UNDER STOCHASTIC LOADINGS

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1 ASME 2013 IDETC/CIE 2013 Paper number: DETC A DESIGN ORIENTED RELIABILITY METHODOLOGY FOR FATIGUE LIFE UNDER STOCHASTIC LOADINGS Zhen Hu, Xiaoping Du Department of Mechanical & Aerospace Engineering Missouri University of Science and Technology

2 Outline Background Fatigue reliability under stochastic loadings Examples Conclusions Future work 2

3 Introduction Structures under stochastic loadings Time dependent reliability Global extreme over time interval Investigated by Mahadevan, Mourelatos, Du, Wang, etc. Fatigue reliability Local extremes Damages accumulation 3

4 Fatigue life analysis Strain-life based method - Related to initial crack Stress-life based method - Based on material S-N curve Fracture mechanics method - Related to initiation and propagation of crack 4

5 Current methods Background Monotonic structure responses to the stochastic loading Assumption of narrow-band or broad-band Based on field stress or strain data Monte Carlo Simulation Efficiency and accuracy may not be good for nonlinear functions w.r.t. stochastic loadings 5

6 Examples Random field loading: Yxt (, ) = µ ( x) + Zxt (, ) = µ ( x) + qj( tφ ) r j= 1 j Nonlinear combination of stochastic processes 6

7 Task St () = g( XY, ()) t - X is a vector of random variables - Y(t) is a vector of stationary stochastic processes p = Pr{ T T} f F Predict fatigue reliability with less computational effort 7

8 Fatigue Reliability Analysis under Stochastic Loadings First Order Reliability Method (FORM) Min u X s.t. DF( T( u X)) 1 Expected accumulated fatigue damage over a time interval conditioned on x = T ( u ) X Peak counting (PC) Level crossing counting (LCC) Range counting (RC) Rainflow counting (RFC) 8

9 Expected fatigue damage analysis The Palmgren-Miner s rule D F ns ( i) ns ( i) = = Ns ( ) 1/( γ s κ ) i= 1 i i i κ DF = v0 T γ s p() s ds - mean upcrossing rate of the stress process - PDF of the stress cycle 9

10 Overview Calculate mean value upcrossing rate Obtain stress cycle distribution from stress response function Determine integration region for damage analysis 10

11 Mean Value Analysis µ S = sf () s ds 0 + µ 1 1 S = F ( P ) 0 S dps 1 r 1 1 ξ 1 1 j + µ s = F ( P ) 0 s dps = wjf 2 j= 1 2 PS = Fs () - CDF of stress response S - Transform stress region to probability region (0,1) 11

12 Mean Value Analysis ξ + 1 Y( t) 2 s.t. ξ j ξ + j + Φ, if uy() t = ξ 1 1 j + Φ, otherwise 2 1 j Max s= F = gt ( ( ux), T( uy() t)) u Inverse MPP search is performed at each of the Gaussian quadrature points 12

13 Mean upcrossing rate analysis Rice s formula: ( ) ( ) ( ) v0 = ωµ t φ βµ () t Ψ βµ ()/ t ωµ () t The mean upcrossing rate is equivalent to the upcrossing rate of an Equivalent Gaussian process Derivation based on the Rice s formula got the similar expression as that from other methods Similar researches have been conducted by Mourelatos, Du, etc. 13

14 Stress cycle distribution analysis For each peak larger than s, mathematically we have: St () > s S( t) = 0 S( t) < 0 f ( St ( ), St ( ), St ( )) S,S,S St () = g( XY, ()) t Computationally expensive 14

15 Stress cycle distribution analysis Z( t) = gt ( ( u ), T( U ( t))) s FORM T Lt () = α () tu () t i X Y i i Y Pr{ Z ( t) = g( u, T( U ( t))) s < 0} i X Y i Pr{ Lt ( ) < βi} =Φ( β i), if si < µ s = Pr{ Lt ( ) < βi} =Φ( βi), otherwise Pr{ Lp( t) < β i}, if si < µ s Pr{ Zp( t) < si} = Pr{ Lp( t) < βi}, otherwise 15

16 Stress cycle distribution analysis For a Gaussian stochastic process with regularity factor η: η=1 0.5 PDF Increase η=0 η ξ 16

17 Stress cycle distribution analysis CDF: FP ( ξ) 2 ξ ξ ηξ =Φ Φ 1 η 1 η 2 ηe 2 2 PDF: fp 2 ξ ξ ηξ ( ξ ) = 1 η φ + ηξ Φ 1 η 1 η 2 2 e 2 2 Regularity factor: ω () t () t C (, t t) () t T 0 12 η = = ω p C t = t = t () t () t ( t, t ) () t Corresponding to the probability of one stress level si 1 2 T 17

18 Stress cycle distribution analysis s 1 1 κ ( ) = ( ( s )) P p γ κ s p s ds γ P P dp µ s 1 P µ s 1 P = 2 µ s 1 κ ( ( s )) γ P P dp p r µ s 1 κ wiγ ( P ( Ppi )) i= 1 s p p 1 Max sp = P ( Ppi ) = g( ux, uy () t, t) u Y( t) s.t. uy() t αi () t = uy() t T ω0 () t i() t C 12 (, t t) i() t ηi = = ωm () t i() t C1122 ( t1, t2 ) i() t t1= t2= t 1 βobj = FP ( Ppi ) η = ηi uy () t = βobj T Transform stress peak region to the probability region Inverse peak distribution analysis is applied 18

19 Mathematical Example S() t = g( XY, ()) t = Y () t + Y () t + Y () t ρ t t = t t Y 1 2 ( 1, 2) exp[ ( 2 1) ] ρy ( t, t ) = cos[ π( t t )] Y ( t) = exp[0.2 U ( t) + 1.5] 3 3 ρ t t = t t U 2 2 ( 1, 2) exp[ ( 2 1) / 0.5 ] 19

20 Results MCS Proposed MCS Proposed PDF 0.2 CDF Peaks of g Peaks of g 20

21 Fatigue Reliability of A Shaft l F(t) d Q(t) 21

22 Results 3 x Stress (Pa) t (hour) 22

23 Conclusions The method is for fatigue reliability under stochastic loadings. Stationary process assumption is made. Peak counting method is employed. Both Gaussian and non-gaussian stochastic loadings can be addressed. 23

24 Future work Consider non-stationary stochastic loadings. Study the other cycle counting methods, such as RC, LCC and RFC. Further improve the efficiency. 24

25 Acknowledgments ONR N (application) NSF CMMI (methodology development) Intelligent Systems Center at Missouri S&T (partial financial support) 25