Fatigue Life Prediction Based on Variable Amplitude Tests

Transcription

1 3 Gaussskt (R=0) Lnljärt (R=0) Hällered (R= 1) F eq / Ekvvalent vdd [kn] 2 1 N = 7.11e+006 S N / Antalet cykler tll brott Thomas Svensson Jacques de Maré Acknowledgements Atlas Copco, Bombarder, Sandvk, Volvo PV, SP, STM, Smögen Workshops

2 Outlne What s metal fatgue? Lfe Predkton & Testng Tradtonal Method Lfe Predkton & Testng Proposed Method Examples Extensons of the model Conclusons 2

3 What s Fatgue? Fatgue s the phenomenon that a materal gradually deterorates when t s subjected to repeated loadngs. Wöhler (1858), Ralway engneer Model for fatgue lfe: Wöhler-curve number of cycles N to falure as functon of cycle ampltude S. log(s) fnte lfe fatgue lmt nfnte lfe log(n) 3

4 Fatgue Lfe, Load Analyss, and Damage SN-curve: N = αs β, materal parameters. 2S Cycle countng Convert a complcated load functon to equvalent load cycles. Load X(t) gves ampltudes S 1, S 2, S 3, Palmlgren-Mner damage accumulaton rule Each cycle of ampltude S uses a fracton 1/N of the total lfe. Damage tme [0,T]: D α 1 Falure occurs when all lfe s used,.e when D>1. T = 1 N = S β tme 4

5 Lfe Predkton & Testng Tradtonal Method Fatgue testng CA Constant Ampltude Analyss Basqun N = αs β 2 Wöhler-curve Predcted lfe VA Varable Ampltude Predkton Palmgren-Mner Ranflow count Basqun S N Dsadvantages: Often systematc predcton errors. Model errors! Could depend on sequence effects, resdual stresses, and threshold effects. Emprcal correcton: Change damage crteron, e.g. D = 1 D = 0.3 5

6 Example: Agerskov Data Set Estmated SN curve from constant ampltude tests 500 Constant ampltude Narrow Broad PM mod Welded steel, non load carryng weld. S eq / Equvalent ampltude [MPa] N = 7.23e+012 S N / Number of cycles to falure Estmaton from CA: Lnear regresson. Here t gves non-conservatve predctons. Soluton suggested by Schütz (1972): Relatve Mner-rule. Calculate a correcton factor based on VA tests,.e. change but keep fxes. 6

7 Lfe Predkton & Testng Proposed Methodology Fatgue testng VA Varable Ampltude Analyss Palmgren-Mner Basqun 2 Wöhler-curve Predcted lfe Predkton 1 VA Palmgren-Mner Varable Ampltude Ranflow count 7 N Basqun S Advantage: Same load type at both testng and predcton. Should reduce possble model errors, and gve better predctons. Inspraton: - Gassner-lne (1950) - Relatv Mner (1972) - Omerspahc (1999) 7

8 Problem Descrpton Model β SN-curve, Basqun: N = αs e A VA load s specfed through a load spectrum,.e. the frequences j of the load ampltudes s j, L=(s j, j ) An equvalent load ampltude s defned as Equvalent Ampltude: S eq ν s β β = j j Damage equvalent to load spectrum. (Palmgren-Mner damage accumulaton) Estmaton Maxmum Lkelhood non-lnear regresson. Uncertanty n estmates. Uncertanty n predcton. v S log(s eq ) Load spectrum s j log(n) Basqun curve log(n) 8

9 ψ = ν Estmaton of Wöhler curve for varable ampltude loads Condense the VA load to a spectrum of counted load cycles and defne ts equvalent load ampltude {, ; = 1,2, m } k s k m S eq = ν S = 1 α Formulate the logarthm of the Basqun equaton... ψ S eq N = S β e β a 1/ β ln N ln β ln Seq, ( ψ ; α β ) = α + ε = f, + ε ε = ln e N(0, σ 2 ) and ML estmate of the parameters from n reference spectrum tests ( ˆ, α ˆ) β = arg mn ˆ σ = s = ( α, β ) 1 n 2 n = 1 [ ( )] 2 ln N f ψ ; α, β n [ ( )] 2 ln N f ψ ; ˆ, α β 2 2 ˆ = 1 9

10 Example: Agerskov Estmated SN-curve from CA-tests, predctons for VA. 500 Estmated SN curve from constant ampltude tests Constant ampltude Narrow Broad PM mod Estmated medan lfe. Confdence nterval for medan lfe. S eq / Equvalent ampltude [MPa] N = 7.23e+012 S % ntervals N / Number of cycles to falure Predcton nterval (for future tests). Relatve lfe, N rel =N/N pred, a way to examne systematc predcton errors. Non-conservatve predctons. Broad: N rel = 0.38; (0.31, 0.47) Narrow: N rel = 0.53; (0.41, 0.69) PM mod: N rel = 0.46; (0.37, 0.59) 10

11 Example: Agerskov Estmated SN-curve from Broad, predcton for CA. Estmated SN curve from varable ampltude tests (Broad) 500 Constant ampltude Broad Estmated medan lfe. Predcton nterval. S eq / Equvalent ampltude [MPa] % ntervals. Conservatve predctons. N rel = 2.57; (1.82, 3.63) No statstcally sgnfcant dfference for the damage exponent. N = 1.49e+012 S N / Number of cycles to falure 11

12 Example: Agerskov Estmated SN-curve from Broad, predcton for Narrow. Estmated SN curve from varable ampltude tests (Broad) 500 Narrow Broad Estmated medan lfe. Predcton nterval. S eq / Equvalent ampltude [MPa] % ntervals. No statstcally sgnfcant systematc errors. N rel = 1.42; (0.98, 2.05) Predcton based on CA gves systematc errors. N rel = 0.53; (0.41, 0.69) N = 1.49e+012 S 2.96 Predcton for PM mod: N rel = 1.23; (0.86, 1.74) N / Number of cycles to falure 12

13 Estmated SN curve Example: Agerskov Estmated SN-curve from all VA, predcton for CA. 500 Constant ampltude Narrow Broad PM mod Estmated medan lfe. Predcton nterval. S eq / Equvalent ampltude [MPa] % ntervals. Possble to combne dfferent types of load spectra when estmatng the SN-curve. Systematc predcton errors. N rel = 2.11; (1.69, 2.64) N = 1.09e+012 S 2.87 No dfference seen n! N / Number of cycles to falure 13

14 Extended Fatgue Model Mean Value Influence Mean value correcton Include mean value correcton when calculatng S eq. Use exstng models, e.g. mean-stress-sensblty (Schütz, 1967) ( s + M s ) β S eq = ν j a, j m, j Possble to estmate the correcton M together wth SN-curve. β Crack closure models Include crack closure when calculatng S eq. ( S S ) β S eq = ν j max, j op, j Use exstng models varable closure level, or constant closure level. Possble to estmate a constant level S op together wth SN-curve. β 14

15 SP: Convex spectrum Mean stress correcton 200 Mean Stress Sensblty (MSS) Convex 1 Convex 0.5 Mean-Stress-Sensblty S a =S a +MS m S eq / Equvalent ampltude [MPa] 100 Uncertanty n parameters = 5.74; 4.74 < < 6.75 (95%) M = 0.17; < M < (95%) Scatter s reduced from s=0.51 to s= N = 2.06e+017 S 5.74 s = , M = N / Number of cycles to falure We should nclude the extra parameter M. Optmal model complexty? 15

16 Conclusons Estmaton of SN-curve from spectrum tests. Methodology based on equvalent load, S eq. Can combne dfferent types of load spectra. Analyss of uncertantes n estmates and predctons. Possble to dstngush systematc devances form random varatons. Dfference between CA and VA? Yes, often! (same ) Dfference between load spectra? No, for Agerskov! Influence from mean value, rregularty, or level crossngs? Extensons Mean value nfluence. Further work. Combned estmaton of SN-curves for CA and for VA (same slope ). Estmate two SN-curves (CA and VA) at the same tme, wth some common parameters. Parameters: ( CA, VA,, ) dfferent Parameters: ( CA, VA,, CA, VA ) dfferentand Dfferent classes of servce loads. same slope,dfferent? 16