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www.cesos.ntnu.no Author – Centre for Ships and Ocean Structureswww.cesos.ntnu.no Gao & Moan – Centre for Ships and Ocean Structures
Frequency-domain multi-modal formulation for fatigue analysis of Gaussian and non-
Gaussian wide-band processes
Dr. Zhen GaoProf. Torgeir Moan
Centre for Ships and Ocean Structures, Norwegian University of Science and Technology
February 24, 2010
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• Introduction• Accuracy of the narrow-band fatigue approximation• Bimodal fatigue analysis• Multi-modal fatigue formulation• Application of non-Gaussian bimodal fatigue analysis to
mooring line tension• Conclusions• Recommendations for future work
Contents
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• Cycle-counting methods for fatigue analysis- frequency-domain methods - time-domain methods
• The narrow-band approximation• Methods for a bimodal (or multi-modal) Gaussian process
- Jiao & Moan (1990) - Lotsberg (2005)- Sakai & Okamura (1995) - Huang & Moan (2006)- Fu & Cebon (2000) - Gao & Moan (2008)
- Olagnon & Guede (2008)• Methods for a general wide-band Gaussian process
- Wirsching & Light (1980) - Zhao & Baker (1992)- Dirlik (1985) - Bouyssy (1993, review paper)- Larsen & Lutes (1991) - Benasciutti & Tovo (2005)
• Non-Gaussian processes- NB Transformation using the high-order moments (e.g. skewness, kurtosis) Winterstein (1988); Sarkani et al. (1994)
Introduction
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Accuracy of the narrow-band fatigue approximation (1)• Spectrum type: multi-modal, Dirlik (1985), Benasciutti and Tovo (2005),
linear wave-induced responses of offshore structures• Total number: around 4200• Vanmarcke’s parameter : 0.038(NB)~0.985(WB)
21
( )G
21 2
2
( )G
21
232
2
21 3
Bimodal (left) and trimodal (right) spectra
Benasciutti and Tovo (2005)omega (rad/s)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Nor
mal
ized
RA
O
0.0
0.2
0.4
0.6
0.8
1.0
1.2Gravity platformFPSOSemi-submersibleTLP
omega (rad/s)0.0 0.5 1.0 1.5 2.0 2.5 3.0
Dou
bly-
peak
ed w
ave
spec
trum
0
2
4
6
8
10
12Hs=6m, Tp=6sHs=6m, Tp=10sHs=6m, Tp=14s
Transfer function (left) and wave spectrum (right)
21 0 21 / /m m m
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• Fatigue damage• Time-domain simulation
– The rainflow counting method is used for comparison! (WAFO)
• Ratio of the NB result to the time-domain result (m=3)
Vanmarcke's bandwidth parameter0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Rat
io o
f nar
row
-ban
d fa
tigue
dam
age
to R
FC
0
1
2
3
4
5
6
7
8
9
10
11
12
13
Vanmarcke's bandwidth parameter0.0 0.1 0.2 0.3 0.4 0.5 0.6
Rat
io o
f nar
row
-ban
d fa
tigue
dam
age
to R
FC
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
The NB approximation is too conservative for these spectra!
Maximum10% over-estimation
Maximum 30% over-estimation
Larger variation
Accuracy of the narrow-band fatigue approximation (2)
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• Some results of linear wave-induced responses– Mudline shear force of a gravity platform– Tension induced by the vertical motion of a TLP– Vertical mid-ship bending moment of a FPSO– Stresses in a brace-column joint of a semi-submersible
Accuracy of the narrow-band fatigue approximation (3)
Vanmarcke's bandwidth parameter0.0 0.1 0.2 0.3 0.4 0.5 0.6
Rat
io o
f fat
igue
dam
age
to R
FC
0.0
0.5
1.0
1.5
SSNBDKBTProposed
Accuracy of the freq.-d. method for fatigue analysis of wave-induced responses
o m e g a ( ra d /s )0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0
Nor
mal
ized
RAO
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
1 .2G ra v ity p la t fo rmF P S OS e m i-s u b m e rs ib leT L P
o m e g a ( ra d /s )0 .0 0 .5 1 .0 1 .5 2 .0 2 .5 3 .0
Dou
bly-
peak
ed w
ave
spec
trum
0
2
4
6
8
1 0
1 2H s = 6 m , T p = 6 sH s = 6 m , T p = 1 0 sH s = 6 m , T p = 1 4 s
Wave spectrum (up); Transfer function (down)
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Bimodal fatigue analysis (1)• Fatigue due to individual components
• Bimodal fatigue problem
• Under the Gaussian assumption (Jiao & Moan, 1990)
• About - Assume that has similar periods as- Time-derivative (Gaussian)- Analytical formula for- Amplitude distribution (Rayleigh sum)- Closed-form solution for when
( ) ( ) ( )HF LFX t X t X t HF LF True NBD D D D
0max
0
2 ( )m mTD y f y dyK
the mean zero up-crossing rate0
max ( )f y the amplitude distribution
HF PD D D is the envelope process of ( ) ( ) ( )HF LFP t R t X t Define ( )HFR t ( )HFX t
( )P t
22 2 2 2 20 0 0* * 1 * *LF LF HF LF HF LF HF LF HF
P HF LFR R R m
EPS 2 2LF HF
( )HFR t ( )LFX t( ) ( ) ( )HF LFP t R t X t
0P
the integralm
ES
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Bimodal fatigue analysis (2)
Vanmarcke's bandwidth parameter0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Rat
io o
f fat
igue
dam
age
to R
FC
0.0
0.5
1.0
1.5
2.0
SSNBDKBTProposed
• Comparison with the rainflow counting method
w1 w2
var1 var2
Spectral density function
SS – Summation of components
NB – Narrow-band approximation
DK – Dirlik’s formula
BT – Benasciutti & Tovo’s formula
Accuracy of the freq.-d. method for bimodal fatigue analysis
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Multi-modal fatigue formulation (1)• Generalization
– Assume the NB components with decreasing central frequencies as
– Define the equivalent processes as– Approximate the fatigue damage as the sum of
1 2( ), ( ),..., ( )iX t X t X t
1 2( ), ( ),..., ( )iP t P t P t
1 2, ,...,P P PiD D D
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Multi-modal fatigue formulation (2)
• Solution for– The Rice formula – Analytical when – Numerical
• Solution for– Rayleigh sum distribution– Analytical when (Narrow-band solution)– Numerical
• Hermite integration method– Convolution integral– Accuracy – Semi-analytical solution
0Pi
1, 2,3i
1i
max ( )Pif y
0 0
1(0, ) (0)2Pi i i i PiPiPi Pip f p dp f
& && & &
2
1* ( ) * ( )
nz
i ii
e f z dz a f z
1 1( ) ( ) ... ( ) ( )i i iP t R t R t X t
1 1( ... )Pi i iR R R R
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Multi-modal fatigue formulation (3)• Comparison with the rainflow counting method
Vanmarcke's bandwidth parameter0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Rat
io o
f fat
igue
dam
age
to R
FC
0.0
0.5
1.0
1.5
2.0
SSNBDKBTProposedVIV and WF+LF
var1
var2 var3
Spectral density function
SS – Summation of components
NB – Narrow-band approximation
DK – Dirlik’s formula
BT – Benasciutti & Tovo’s formula
VIV and WF+LF – Summation of the VIV fatigue and the combined WF and LF fatigue
1 2 3
Accuracy of the freq.-d. method for trimodal fatigue analysis
FD
RFC
DD
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var1
w1
var2 var3
w2 w3
var1=var2=var3
Multi-modal fatigue formulation (4)• General wide-band Gaussian processes• Basic idea
– Discretize the wide-band spectrum into three segments– Approximate each segment narrow-banded– Obtain the fatigue damage as for a trimodal process
• Considerations– Which rule to discretize? (numerically accurate / efficient?)– How good the NB approximation is for each segment? (especially for
high frequencies? number of segments?)
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Multi-modal fatigue formulation (5)
Vanmarcke's bandwidth parameter0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Rat
io o
f fat
igue
dam
age
to R
FC
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5SSNBDKBTProposed
Spectral density function (Benasciutti & Tovo, 2005) Accuracy of the freq.-d. method for general wide-band
fatigue analysis
FD
RFC
DD
• Case study of generally defined wide-band spectra
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Application of non-Gaussian bimodal fatigue analysis to mooring line tension (1)• Mooring system analysis:
• Sources of nonlinearity:– Second-order wave forces acting on vessel– Drag force acting on mooring lines– Nonlinear offset-tension curve
• The Gaussian assumption is made in current design codes for mooring systems.- ISO 19901-7 (2005) - API RP 2SK (2005) - DNV OS-E301 (2004)
Wind
Wave
Current
OriginalPosition
MeanPosition
Dynamic Analysis (WF+LF)
Static Analysis
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• Mooring line tension in a stationary sea state
– Pre- and mean tension due to steady wind, wave and current forces (time-invariant)
– Wave-frequency (WF) line tension (dynamic, short period (e.g. 10-15 sec)), skewness=0, kurtosis=3
– Low-frequency (LF) line tension (quasi-static, long period (e.g. 1-2 min)), skewness=0.8, kurtosis=4.5
Application of non-Gaussian bimodal fatigue analysis to mooring line tension (2)
( ) ( ) ( )P M WF LFT t T T T t T t
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• WF mooring line tension (Morison formula)
• Amplitude distribution(Borgman, 1965)
Application of non-Gaussian bimodal fatigue analysis to mooring line tension (3)
y0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
f(y)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0k=0.2k=0.5k=1k=2
( ) ( ) ( ) ( )WF d mT t k u t u t k u t 2
d u
m u
kk
k
Amplitude distribution of WF tension
2
2 2
2 2
0
max
(3 1) exp( (3 1) )2
3 1 3 1exp( ( ))
2 2 2
( )WFT
yk y k
yk ky
k k
yf
0
0
0
,
y y
y y
2
0 1 / (2 3 1)y k k where
Drag dominant (Exponential)
Inertia dominant (Rayleigh)
where
Combined Rayleigh and exponential distribution!
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Application of non-Gaussian bimodal fatigue analysis to mooring line tension (4)
Distributions of the fundamental variablesx
-5 -4 -3 -2 -1 0 1 2 3 4 5
f X(x
)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8Standard Gaussian variableScaled 2nd-order forceScaled LF vessel motionScaled LF mooring line tension
• LF mooring line tension
• LF forces, motions and time-derivatives– Second-order Volterra series (Næss, 1986)– Sum of exponential distributions
• LF tension and time-derivative– Transformation (offset-tension)
• Amplitude distribution– The Rice formula (Rice, 1945)
Second-order wave forces
Linearized model LF vessel motions
Offset-tension curve LF line tension
,0( ) ( , )Tlf Tlf Tlfy yf y y dy
max
( )1( )(0)
TlfTlf
Tlf
d yf y
dy
Skewness>0 Kurtosis>3
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Hs (m)3 4 5 6 7 8
Rel
ativ
e di
ffere
nce
of fa
tigue
dam
age
-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
Tp=7.5sTp=9.5sTp=11.5sTp=13.5s
• Comparison of the frequency-domain method for fatigue analysis with time-domain simulations (Gao & Moan, 2007)
– Short-term sea states:Hs=3.25 - 7.75mTp=7.5 - 13.5sec
– Accuracy:WF: -13% - 2%LF: -3% - 12%Comb.: -10% - 11%
Accuracy of the freq.-d. method for fatigue analysis
Black – WF; Red – LF; Green – Combined fatigueFD RFC
RFC
D DD
Application of non-Gaussian bimodal fatigue analysis to mooring line tension (5)
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• Depending on the bandwidth parameter, the narrow-band fatigue approximation might be still applicable to some linear wave-induced structural responses in ocean engineering.
• For a general wide-band Gaussian process, the formulae given by Dirlik and Benasciutti & Tovo gives accurate estimation of fatigue damage.
• The multi-modal fatigue formulation method, including the bimodal one, predicts accurately the fatigue damage of ideal Gaussian processes with multiple peaks. It can also be applied to non-Gaussian processes.
Conclusions
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• Application in design code– Mooring system (ISO 19901-7, API RP 2SK, DNV OS-E301)– Formulae by Dirlik and Benasciutti & Tovo might be used for general wide-band
Gaussian processes
• Fatigue of non-Gaussian processes– Definition by e.g. distributions or statistical moments– Effect of bandwidth and non-Gaussianity
• Other application of the existing methods
Recommendations for future work
* *NG NB G NBFD FD
Spectra of overturning moment
An example of multi-modal response of
offshore fixed wind turbines
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[1] Jiao, G. & Moan, T. (1990) Probabilistic analysis of fatigue due to Gaussian load processes. Probabilistic Engineering Mechanics; Vol. 5, No. 2, pp. 76-83.
[2] Sakai, S. & Okamura, H. (1995) On the distribution of rainflow range for Gaussian random processes with bimodal PSD. The Japan Society of Mechanical Engineers, International Journal Series A; Vol. 38, No. 4, pp. 440-445.
[3] Fu, T.T. & Cebon, D. (2000) Predicting fatigue lives for bi-modal stress spectral densities. International Journal of Fatigue; Vol. 22, pp. 11-21.
[4] Lotsberg, I. (2005) Background for revision of DNV-RP-C203 fatigue analysis of offshore steel structure. Proceedings of the 24th International Conference on Offshore Mechanics and Arctic Engineering, Halkidiki, Greece; Paper No. OMAE2005-67549.
[5] Huang, W. & Moan, T. (2006) Fatigue under combined high and low frequency loads. Proceedings of the 25th International Conference on Offshore Mechanics and Arctic Engineering, Hamburg, Germany; Paper No. OMAE2006-92247.
[6] Gao, Z. & Moan, T. (2008) Frequency-domain fatigue analysis of wide-band stationary Gaussian processes using a trimodal spectral formulation. International Journal of Fatigue; Vol. 30, No. 10-11, pp. 1944-1955.
[7] Olagnon, M. & Guede, Z. (2008) Rainflow fatigue analysis for loads with multimodal power spectral densities. Marine Structures; Vol. 21, pp. 160-176.
[8] Wirsching, P.H. & Light, M.C. (1980) Fatigue under wide band random stresses. Proceedings of the American Society of Civil Engineers, Journal of the Structural Division; Vol. 106, No. ST7, pp. 1593-1607.
[9] Dirlik, T. (1985) Application of computers in fatigue. Ph.D. Thesis, University of Warwick.[10] Larsen, C.E. & Lutes, L.D. (1991) Predicting the fatigue life of offshore structures by the single-
moment spectral method. Probabilistic Engineering Mechanics; Vol. 6, No. 2, pp. 96-108.
References (1)
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www.cesos.ntnu.no Author – Centre for Ships and Ocean Structureswww.cesos.ntnu.no Gao & Moan – Centre for Ships and Ocean Structures
[11] Zhao. W. & Baker, M.J. (1992) On the probability density function of rainflow stress range for stationary Gaussian processes. International Journal of Fatigue; Vol. 14, No. 2, pp. 121-135.
[12] Bouyssy, V., Naboishikov, S.M. & Rackwitz, R. (1993) Comparison of analytical counting methods for Gaussian processes. Structural Safety; Vol. 12, pp. 35-57.
[13] Benasciutti, D. & Tovo, R. (2005) Spectral methods for lifetime prediction under wide-band stationary random processes. International Journal of Fatigue; Vol. 27, pp. 867-877.
[14] Winterstein S.R. (1988) Nonlinear vibration models for extremes and fatigue. American Society of Civil Engineers, Journal of Engineering Mechanics; Vol. 114, No. 10, pp. 1772-1790.
[15] Sarkani, S., Kihl, D.P. & Beach, J.E. (1994) Fatigue of welded joints under narrow-band non-Gaussian loadings. Probabilistic Engineering Mechanics; Vol. 9, pp. 179-190.
[16] ISO (2005) Petroleum and natural gas industries - Specific requirements for offshore structures -Part 7: Stationkeeping systems for floating offshore structures and mobile offshore units. ISO 19901-7.
[17] API (2005) Recommended practice for design and analysis of stationkeeping systems for floating structures. API RP 2SK.
[18] DNV (2004) Offshore Standard - Position Mooring. DNV OS-E301.[19] Borgman L.E. (1965) Wave forces on piling for narrow-band spectra. Journal of the Waterways
and Harbors Division, ASCE; pp. 65-90.[20] Næss, A. (1986) The statistical distribution of second-order slowly-varying forces and motions.
Applied Ocean Research; Vol. 8, No. 2, pp. 110-118.[21] Gao, Z. & Moan, T. (2007) Fatigue damage induced by non-Gaussian bimodal wave loading in
mooring lines. Applied Ocean Research; Vol.29, pp. 45-54.
References (2)
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Thank you!