1. FAST SPECTRAL RAINFLOW FATIGUE DAMAGE ASSESSMENT UNDER WIDEBAND MULTIPEAK LOADING Michel OLAGNON & Zakoua GUÉDÉ IFREMER, Brest (France) Fatigue Workshop - 24 th February 2010 Broadband spectral fatigue: from gaussian to non-gaussian, from research to industry

2. Context Fine description of sea states climate as partition into waves systems Complex fatigue damage assessment a large set of operational sea states (combinations of all the wave systems components) has to be considered 1

3. metocean database discretization + statistics scatter diagram (jPDF of H S, T p,  ) dimension = 3 10 2 to 10 3 fatigue loadings unimodal loadings metocean database discretization + statistics scatter diagrams (jPDF of H S (i), T p (i),  (i), i=1,2,3) dimension = 9 10 6 to 10 9 fatigue loadings. partition wave systems data (swells, wind sea) multimodal loadings 2

4. Objective Use the Iterative Components Addition (ICA) formulas set up earlier [Olagnon & Guédé, 2008] to simplify the damage computation. ICA formulas allow to compute the damage of a multimodal loading spectra in terms of the damage of its individual components taken separately, keeping a low level of conservatism. D( S1 + S2 + S3 … ) = ICA( D(S1), D(S2), D(S3) …) Full damage computation [Stress – Rainflow – Miner] needed only for the wave systems components and ICA-based damage computation for the large set of all their combinations by ICA formula. drastic reduction of computation time 3

5. Outline 1. ICA FROMULA 2. APPLICATION: FPSO ON WEST AFRICA AREA 3. CONCLUSION & PERSSPECTIVES 4

6. BASIC IDEA UNDER ICA FORMULA sum signal LF signal Max A ; min A Max B ; min B Partition of the set of turning points  LF narrow band  Miner damage with S-N curve: N = K S m  HF & LF clearly separated Assumptions From the mathematical formulation of the rainflow counting [Rychlik, 1987], each subset is stable by rainflow counting D S = D A + D B 1. ICA FORMULA 5

7. EXPRESSION OF DAMAGES D A & D B 1. ICA FORMULA factor due to reduction in number of cycles scaling effect due to addition of low- frequency signal (NB) ( N B = N LF ) E[M B m ] & E[M B ] are estimated with a slightly conservative approximation of the distribution of M B in terms of some spectral moments 6

8. 1. ICA FORMULA Approximation of M B distribution The local maxima of the composite signal follow a Rice distribution [1945], N S /N LH = 1/  : mean number of peaks between two successive zero- upcrossings. The peaks are assumed independent (conservative assumption) sum signal LF signal Max A ; min A Max B ; min B M B is the maximum among the local maxima between two successive zero upcrossings of the low-frequency signal 7

9. 1. ICA FORMULA Conservatism level The conservatism level (Clevel) is higher for larger  and lower  For m = 3, Clevel > 130% for  ;  and  ;  For m = 5, Clevel > 130% for  8 0.1    m h,0 /m t,0  0.9 ; 1.2   T p,h /T p;l  20 with T p,h = 5s + m = 3 + m = 5  130%

10. Briefly,  ICA is simple and depends only on the main spectral properties of the signal components, namely their variance, irregularity factor and zero- crossing period (m 0,m 2,m 4 ).  ICA can be used recursively when the signal has more than two modes (no narrow-band assumption on high-frequency component). 1 2 3  ICA becomes inconvenient with two-slopes S-N curve. Nevertheless, a formula which allows to obtain the two-slopes S-N damage in terms of the simple S-N has been set up (Olagnon, 2009 – in submission). 1. ICA FORMULA 9

11. Industrial application  Wave loadings: from metocean database of measured sea states in the West Africa area (almost 3 years)  Mechanical modeling: Wave bending moment of an FPSO Hull girder  linear hydrodynamic response (with RAO)  Fatigue design requirement: double-slopes S-N curve from Bureau Veritas requirements, 100 years design lifetime Work  Assessment of the occurrence probabilities of all the operational sea states given under some assumptions  Assessment of the total damage from the metocean database with time-domain simulations from the operator’s specifications with a ICA-based method 2. APPLICATION 10

12. Metocean database partition & assumptions on metocean climate  Assumptions made for the re-construction of the metocean climate (H1) : independent wave systems (H2) : wave systems conditioned on the type of combination Including a requirement to meet the proportions observed in the database and exclusion criteria on the ratio of the peak periods and the discrepancy between the directions (e.g. close wave systems components, unrealistic combinations) 2. APPLICATION  Partitions of the sea states of the metocean database 11

13. 12 2. APPLICATION Joint occurrence probabilities 1. Directional scatter diagrams of the Wave systems Suit the discretization to the criteria: log-scale (multiplicative) classes for the periods half criterion threshold as class width for directions and the log(period). 2. Joint occurrence probabilities of all the possible combinations Given under independence assumption and truncated to the set of possible combinations. product of the wave systems frequencies from the scatter diagrams normalized by the sum of the frequencies of the possible combinations As a result, observed MS fall into 169 classes, SS into 148, WS into 165. (H1) :  2 millions combinations allowed to occur (H2) :  800 thousand combinations allowed to occur

14. Comparison of statistical properties of H S under assumption (H1) metocean database re-built metocean climate 13 2. APPLICATION Significant discrepancies between statistics derived from the hypothesis (H1) and that of the database. The fact that the H S are higher for MS only is not reflected by (H1).

15. Smaller discrepancies between statistics derived from the assumption (H2) and that of the database. However, under (H2) the H S are greater for combinations without SS and lower for combinations with SS. Both assumptions do not reflect the trends observed in the metocean database. 14 2. APPLICATION Comparison of statistical properties of H S under assumption (H2) metocean database re-built metocean climate

16. ICA-BASED DAMAGE ASSESSMENT PROCEDURE 1.Partition & Discretization of the wave spectra of the metocean database 2.Determination of the wave systems combinations probabilities of occurrence 3.Calculation of the responses spectra to the wave systems components 4.Responses partition and identification of the « response systems » 5.Calculation of the damages and other characteristics (spectral moments) of the response systems 6.Combination within each possible sea state (i.e. each wave systems combination) using ICA formula 7.Summation with the probabilities of occurrence 15 2. APPLICATION

17. Fatigue damage from metocean database (Reference) * Partition with « triangle » spectral shape model 16 2. APPLICATION  The sea states with higher H S are responsible for a greater proportion of the damage. D HS,1/3 represents 95% of the damage and D HS,1/10 represents 70%. But this result depends on the structural response under consideration.  The triangle spectral shape considered here (which does not take into account the tail of the spectra) yields a non-gaussian process and gives lower damages. A « wallops » spectral shape model (with a high frequency tail) would provide larger results.

18.  The damage computed with the ICA-based procedure suit the effects of the assumptions made for the metocean climate re- construction.  Concerning the computation efficiency. The step n°7 of the ICA- based procedure last some seconds with a FORTRAN program on a fast computer (  15s for 2 millions combinations). 17 2. APPLICATION Fatigue damage from ICA-based assessment method * Partition with « triangle » spectral shape model

19. CONCLUSION  ICA formula is simply implemented in a general procedure for fatigue damage assessment under multimodal sea states with good performance;  This formula provided reasonably conservative results for the actual application. Problems highlighted:  The use of two-slopes S-N curve. Problem solved using a formula to obtain the damage with double slope S-N curve from those with single slope S-N curve.  the metocean climate constructed under the independence assumption is not satisfactory. (H1) shows major discrepancies with the database. (H2) is better but not totally satisfactory. 18

20. PERSPECTIVES  In the short term, other applications to be considered :  a very low-frequency response in addition to the wave systems responses  structural responses which are significantly direction-dependent  non-linear responses which can be represented by a linear response corrected with a factor depending on some parameters (e.g. H S,T p )  In a longer term, extend the approximation of the distribution of M B to non-gaussian responses.  Improve the re-construction of the metocean climate, using instead on the overall statistics of environmental parameters, their evolution over time (events statistics). 19

21. Thank you for your attention !!!