The ICA Formula and Double Slope S-N curves 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

The ICA Formulas D = (NH-NL)/NH DH + F(, , m) DL ML distributed as the maximum of NH/Nl maxima of the global signal : D = (1-) DH + Z1(m, ) ((1-2)/(1-2))m/2 DL ML distributed as the highest NH/Nl maxima of the global signal : D = (1-) DH + Z2(m, ) ((1-2)/(1-2))m/2 DL ML distributed as the highest NH/Nl narrow-band maxima of the global signal : D = (1-) DH + Q(m/2+1, -Ln()) / (1-2)m/2 DL DDNB = (1-) DH + (1+  /(1-2))m DL 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

The ICA Formulas D = (NH-NL)/NH DH + F(, , m) DL The amplification of DL by F moves it from below to above the threshold where we change from slope 5 to slope 3 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

The ICA Formulas D+ + D- = (NH-NL)/NH DH + F(, , m) DL There is no robust expression for D+ and D- from the DH and DL counterparts, especially DH+ and DL+ may well be zero and D+ non-zero. Thus the attempt: express D+ = F+(D3, D5) D- = F- (D3, D5) 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

Michel Olagnon, Zakoua Guédé, Hélène Pineau D+ + D- = D5 – D5+ + D3+ The problem is now to find an upper and a lower bound for the damage of ranges above the threshold, or for their distribution. We can use the fact that the largest ranges are zero-crossing, and that rainflow filtering at the threshold is the same as using a [thus narrow-band] “phantom process” (Lindgren & Rydén). We can also use the fact that min-max counting is unconservative and double amplitude conservative w.r.t. rainflow on the phantom process. A factor  that relates crest-trough heights to double crests can be computed from the signal autocorrelation at delay T/2 (Naess). 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

Michel Olagnon, Zakoua Guédé, Hélène Pineau 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

Michel Olagnon, Zakoua Guédé, Hélène Pineau The incomplete moments of a Rayleigh distribution are provided by the regularized lower and upper Gamma functions P and Q. 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

Michel Olagnon, Zakoua Guédé, Hélène Pineau 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

Michel Olagnon, Zakoua Guédé, Hélène Pineau D = D5 P(7/2, s0/ )+ D3 Q(5/2, s0/ ) (a) D = D5 P(7/2, s0/ )+ D3 Q(5/2, s0/ ) (b) D = D5 P(7/2, s0/ )+ D3 Q(5/2, s0) (c) D = D5 P(7/2, s0/ )+ D3 Q(5/2, s0) (d) s0 is the normalized value S0/22 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

Michel Olagnon, Zakoua Guédé, Hélène Pineau 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

Michel Olagnon, Zakoua Guédé, Hélène Pineau 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau

Michel Olagnon, Zakoua Guédé, Hélène Pineau Conclusion The double slope S-N curve is no longer a problem. We can see light at the end of the tunnel(*). (*)or is it a fast train coming our way ? 26/04/2019 Michel Olagnon, Zakoua Guédé, Hélène Pineau