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1

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

2


• 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

3


• 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

4


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)

2D Graph 1

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

2D Graph 2

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

5


• 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

Ra

tio o

f na

rro

w-b

and

fat

igue

da

ma

ge t

o 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 na

rro

w-b

and

fatig

ue d

am

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)

6


• 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)

Fatigue damage ratio to RFC as a function of Vanmarcke's bandwidth parameter

Vanmarcke's bandwidth parameter0.0 0.1 0.2 0.3 0.4 0.5 0.6

Ratio

of

fatig

ue d

am

age t

o R

FC

0.0

0.5

1.0

1.5

SSNBDKBTProposed

Accuracy of the freq.-d. method for fatigue analysis of wave-induced responses

2D Graph 1

omega (rad/s)0.0 0.5 1.0 1.5 2.0 2.5 3.0

Norm

alized R

AO

0.0

0.2

0.4

0.6

0.8

1.0

1.2Gravity platformFPSOSemi-submersibleTLP

2D Graph 2

omega (rad/s)0.0 0.5 1.0 1.5 2.0 2.5 3.0

Do

ub

ly-p

ea

ke

d w

ave

sp

ectr

um

0

2

4

6

8

10

12Hs=6m, Tp=6sHs=6m, Tp=10sHs=6m, Tp=14s

Wave spectrum (up); Transfer function (down)

7


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 dy

K

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

EPS2 2LF HF

( )HFR t ( )LFX t

( ) ( ) ( )HF LFP t R t X t

0P

the integralm

ES

8


Bimodal fatigue analysis (2)

Fatigue damage ratio to RFC as a function of Vanmarcke's bandwidth parameter


Rat

io o

f fa

tigue

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

9


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

10


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 Pi

p f p dp f

& && & &

2

1

* ( ) * ( )n

zi i

i

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

11


Multi-modal fatigue formulation (3)• Comparison with the rainflow counting methodFatigue damage ratio to RFC as a function of Vanmarcke's bandwidth parameter


Rat

io o

f fa

tigue

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

D

D

12


var1

w1

var2 var3

w2w3

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?)

13


Multi-modal fatigue formulation (5)Fatigue damage ratio to RFC as a function of Vanmarcke's bandwidth parameter


Rat

io o

f fa

tigue

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

D

D

• Case study of generally defined wide-band spectra

14


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

15


• 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

16


• WF mooring line tension (Morison formula)

• Amplitude distribution

(Borgman, 1965)


fydistribution

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 m

T 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

01 / (2 3 1)y k k where

Drag dominant (Exponential)

Inertia dominant (Rayleigh)

where

Combined Rayleigh and exponential distribution!

17



Distributions of the fundamental variables

pdf of the variables

x-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)Tlf

TlfTlf

d yf y

dy

Skewness>0 Kurtosis>3

18


Error of Combined WF and LF Fatigue Damage

Hs (m)3 4 5 6 7 8

Rel

ativ

e di

ffer

ence

of

fatig

ue d

amag

e

-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 - 7.75m

Tp=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 fatigue

FD RFC

RFC

D D

D


19


• 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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