Frequency-domain multi-modal formulation for fatigue ... · PDF fileFrequency-domain multi-modal formulation for fatigue analysis of Gaussian and non- ... - ISO 19901-7 (2005) - API

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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

www.cesos.ntnu.no CeSOS – Centre for Ships and Ocean Structures

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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

8


Bimodal fatigue analysis (2)


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


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)


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)


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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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


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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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[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!

Frequency-domain multi-modal formulation for fatigue ... · PDF fileFrequency-domain multi-modal formulation for fatigue analysis of Gaussian and non- ... - ISO 19901-7 (2005) - API

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