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FAST SPECTRAL RAINFLOW FATIGUE DAMAGE
ASSESSMENT UNDER WIDEBAND MULTIPEAK
LOADING
Michel OLAGNON & Zakoua GUÉDÉ
IFREMER, Brest (France)
Fatigue Workshop - 24th February 2010
Broadband spectral fatigue: from gaussian to non-gaussian, from research to industry
Context
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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
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unimodal loadings
metocean database
metocean database
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partition
wave systems data
(swells, wind sea)
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2
multimodal loadings
102
discretization
+
statistics
discretization
+
statistics
scatter diagram (jPDF
of HS, Tp, q)
scatter diagrams (jPDF of
HS(i), Tp(i), q(i) , i=1,2,3)
dimension = 3
to 103 fatigue loadings
106
dimension = 9
to 109 fatigue loadings.
Objective
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Use the Iterative Components Addition (ICA) formulas set up earlier
[Olagnon & Guédé, 2008] to simplify the damage computation.
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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
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Outline
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1.
ICA FROMULA
2.
APPLICATION: FPSO ON WEST AFRICA AREA
3.
CONCLUSION & PERSSPECTIVES
BASIC IDEA UNDER ICA FORMULA
Assumptions
1. ICA FORMULA
Partition of the set of turning points
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 HF & LF clearly separated
sum signal
LF signal
MaxA ; minA
MaxB ; minB
 LF narrow band
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 Miner damage with
S-N curve: N = K Sm
From the mathematical formulation of the rainflow counting
[Rychlik, 1987], each subset is stable by rainflow counting
DS = DA + DB
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1. ICA FORMULA
EXPRESSION OF DAMAGES DA & DB
kN
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DA  k N 
factor due to
reduction in
number of cycles
scaling effect due
to addition of lowfrequency signal
ka 
MA
; M : rangemean value
M HF
from N S M S  N A M A  N LF M B
we obtain: k a  1 
DB 
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 ka m  DHF
NA
N LH

 1  ;  
NS
NS

 M  m 
E  B   DLF
m m

   
1      1
2


1
DLF 
N LF
K
N
DB  LF
K


M B  M HF
1 
MS
2  lf
 2 
m

  m2  1
m
 M  m 
E  B   (NB
   
(NB)
= NLF)
 hf

E[MBm] & E[MB] are estimated with a slightly conservative approximation
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of the distribution of MB in terms of some spectral moments
1. ICA FORMULA
Approximation of MB distribution
MB is the maximum among the local maxima between two successive zero
upcrossings of the low-frequency signal
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sum signal
LF signal
MaxA ; minA
MaxB ; minB
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The local maxima of the composite signal follow a Rice distribution [1945], f , F
NS/NLH = 1/ : mean number of peaks between two successive zeroupcrossings. The peaks are assumed independent (conservative assumption)
Y1 ( x)  F ( x)1  ,
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y1 ( x) 
1

f ( x) F ( x) 1  1
1. ICA FORMULA
Conservatism level
0.1    mh,0/mt,0  0.9 ; 1.2  x  Tp,h/Tp;l  20 with Tp,h = 5s
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x
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+m=3
+m=5
130%
The conservatism level (Clevel) is higher for larger x and lower 
For m = 3, Clevel > 130% for   0.1 ; x > 5 and   0.2 ; x > 7
For m = 5, Clevel > 130% for x > 7.
1. ICA FORMULA
Briefly,
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 ICA is simple and depends only on the main spectral properties of the
signal components, namely their variance, irregularity factor and zerocrossing period (m0,m2,m4).
 ICA can be used recursively when the signal has more than two modes
(no narrow-band assumption on high-frequency component).
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1
 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).
2. APPLICATION
Industrial application
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 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
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• from the metocean database with time-domain simulations
• from the operator’s specifications with a ICA-based method
2. APPLICATION
Metocean database partition & assumptions on
metocean climate
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 Partitions of the sea states of the metocean database
combinations
MS only
MS + SS
MS + WS
MS + SS + WS
no MS
total
number
1212
2740
1536
2549
3
8040
frequency [%]
15,07
34,08
19,1
31,7
0,05
100
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 Assumptions made for the re-construction of the metocean climate
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(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)
Joint occurrence probabilities
2. APPLICATION
1. Directional scatter diagrams of the Wave systems
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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
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Given under independence assumption and truncated to the set of
possible combinations.
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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
2. APPLICATION
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Comparison of statistical properties of HS under
assumption (H1)
stats. prop.
mean HS
std HS
HS,1/3
HS,1/10
MS only
1,59 0,99
0,48 0,49
2,14 1,25
2,63 1,75
MS + SS
1,4 1,27
0,4 0,5
1,85 1,77
2,23 2,15
MS + WS
1,25 1,2
0,32 0,46
1,6 1,46
1,9 1,9
MS + SS + WS
1,22
1,43
0,32
0,47
1,58
1,92
1,91
2,28
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metocean database
re-built metocean climate
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Significant discrepancies between statistics derived from the
hypothesis (H1) and that of the database. The fact that the HS
are higher for MS only is not reflected by (H1).
2. APPLICATION
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Comparison of statistical properties of HS under
assumption (H2)
stats. prop.
mean HS
std HS
HS,1/3
HS,1/10
MS only
1,59 1,64
0,48 0,5
2,14 2,25
2,63 2,75
MS + SS
1,4 1,32
0,4 0,44
1,85 1,77
2,23 2,15
MS + WS
1,25 1,27
0,32 0,36
1,6 1,46
1,9 1,9
MS + SS + WS
1,22
1,15
0,32
0,35
1,58
1,48
1,91
1,79
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metocean database
re-built metocean climate
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Smaller discrepancies between statistics derived from the
assumption (H2) and that of the database. However, under (H2)
the HS are greater for combinations without SS and lower for
combinations with SS.
Both assumptions do not reflect the trends observed in the
metocean database.
2. APPLICATION
ICA-BASED DAMAGE ASSESSMENT PROCEDURE
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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
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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
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2. APPLICATION
Fatigue damage from metocean database
(Reference)
Prob. Occ. [%]
Dt
MS only
MS + SS
MS + WS
MS + SS + WS
total
15,07
34,08
19,01
31,7
100
0,2863
0,2542
0,0604
0,0748
0,6757
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sea states
valeur
0,2684
0,2263
0,0548
0,0613
0,6108
Dt*
Ratio ./Dt
93,75%
89,02%
90,73%
81,95%
90,40%
DHS,1/3
valeur Ratio ./Dt
0,2557 89,31%
0,2314 91,04%
0,0567 93,89%
0,0692 92,57%
0,6358 94,09%
DHS,1/10
valeur Ratio ./Dt
0,1492 52,11%
0,1618 63,66%
0,0438 72,55%
0,0547 73,17%
0,4772 70,62%
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* Partition with « triangle » spectral shape model
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 The sea states with higher HS are responsible for a greater
proportion of the damage. DHS,1/3 represents 95% of the damage
and DHS,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.
2. APPLICATION
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Fatigue damage from ICA-based assessment
method
sea states
Prob. Occ. [%]
Dt*
MS only
MS + SS
MS + WS
MS + SS + WS
total
15,07
34,08
19,01
31,7
100
0,2684
0,2263
0,0548
0,0613
0,6108
DICA - (H1)
valeur Ratio ./Dt*
0,0808 30,11%
0,2483 109,73%
0,1005 183,42%
0,233 380,06%
0,6626 108,49%
DICA - (H2)
valeur Ratio ./Dt*
0,3615 134,70%
0,217 95,91%
0,0721 131,52%
0,0638 104,15%
0,7145 116,98%
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* Partition with « triangle » spectral shape model
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 The damage computed with the ICA-based procedure suit the
effects of the assumptions made for the metocean climate reconstruction.
 Concerning the computation efficiency. The step n°7 of the ICAbased procedure last some seconds with a FORTRAN program on
a fast computer ( 15s for 2 millions combinations).
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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.
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Problems highlighted:
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 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.
PERSPECTIVES
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 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
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 non-linear responses which can be represented by a linear response
corrected with a factor depending on some parameters (e.g. HS,Tp)
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 In a longer term, extend the approximation of the distribution of
MB 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).
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Thank you for your attention !!!