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Fatigue under Bimodal Loads

Zhen Gao

Torgeir Moan

Wenbo Huang

March 23, 2006

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Contents

• Bimodal random process

• Methods for bimodal fatigue damage assessment

• Case study of mooring system of a semi-submersible

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Bimodal random process

• A wide-band process with a bimodal spectral density.

• Examples:– Torque of propellers

(or thrusters) in waves

– Mooring line tension

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Fatigue based on S-N curve and Miner rule• Gaussian narrow-band fatigue damage

• Fatigue damage of a bimodal process

0

0

0( ) ( 2 ) (1 )2

m mS

T T mD s f s ds

K K

where

is the mean zero up-crossing rate.0 is the standard deviation of the process.

( ) ( ) ( )LF HFX t X t X t

LF HF NBD D D D

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Methods for bimodal fatigue (1)• Time domain methods

– Peak counting– Range counting

• Spectral methods for a general wide-band Gaussian process

– Wirsching & Light (1980)– Zhao & Baker (1992)

• Spectral methods for a bimodal Gaussian process– Single moment method– Sakai & Okamura (1995)– DNV formula (2005)

• Non-Gaussian process– Transformation (Winterstein,1988; Sarkani et al.,1994)

– Level crossing counting– Rainflow counting

– Dirlik (1985)– Benasciutti & Tovo (2003)

– Jiao & Moan (1990)– Fu & Cebon (2000)– Huang & Moan (2006)

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Methods for bimodal fatigue (2)• Jiao & Moan (1990)

• DNV (2005)

P HFD D D where is the envelope process of

( ) ( ) ( )LF HFP t X t R t Assume

Then

( )HFR t

( ) ( ) ( )LF HFX t X t X t

( )HFX t

( )mLF HF LFLF HF HF

LF HF

D

nD D D

n

For Gaussian processes, analytical formula can be obtained.

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Methods for bimodal fatigue (3)• Fatigue damage estimation of a Gaussian

bimodal process

Jiao & Moan (1990) DNV (2005)

DNV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0

2

4

6

8

10

12

14

16

18

200.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

v0HF / v0LF

LFLF

WF2

2 2LF

LF WF

0

0

HF

LF

Jian and Moan with Bandwidth 0.1

LFLF

WF

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

v0HF / v0LF

0

2

4

6

8

10

12

14

16

18

200.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

2

2 2LF

LF WF

0

0

HF

LF

NB

D

D

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Case study of mooring system

• A semi-submersibleMain particulars of the semi-submersible

Displacement (ton) 52500

Length O.A. (m) 124

Breadth (m) 95.3

Draught (m) 21

Operational water depth (m) 340

• Mooring system– Line No.10– Pre-tension of 1320 kN– Studless chain link with a dia

meter of 125 mm

Horizontal projection of the mooring system

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Mooring line tension components

• Pre-tension and mean tension due to steady wind, wave and current forces (time-invariant)

• LF line tension (quasi-static, long period (e.g. 1 min))

• WF line tension (dynamic, short period (e.g. 15 sec))

• Both LF and WF tension are narrow-band.

• Bimodal with well-separated low and wave frequencies

• Independent assumption between LF and WF tension

( ) ( ) ( )P M LF WFT t T T T t T t

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Low frequency (LF) line tension

• Distribution of slowly-varying wave force and motion can be expressed by a sum of exponential distributions given by an eigenvalue problem (Næss, 1986)

• The LF line tension can be quasi-statically determined the line characteristic (cubic polynomial, even linear)

• Distribution of the amplitude of LF tension depends on the fundamental tension process and its time-derivative.

1

1

exp( )2 2

( )

exp( )2 2

Mj

j j j

Z Nj

j M j j

l z

f zl z

0

0,

z

z

1

1

(1 )N

kj

k jk j

l

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Wave frequency (WF) line tension (1)

• Simplified dynamic model (Larsen & Sandvik, 1990)

• Distribution of the amplitude of WF line tension (Combined Rayleigh and exponential distribution)(Borgman, 1965)

* 2 *( ) ( ) ( ) ( ) ( )TUEWF G WFT t c u t u t k u t m x t

Basically, it is a Morison formula with a drag term and an equivalent inertia term.

max

22 2

2 20

(3 1) exp( (3 1) )2

( )3 1 3 1

exp( ( ))2 2 2

WFT

yk y k

f yyk k

yk k

0

0

0

,

y y

y y

is a measure of the relative importance of the drag term and the equivalent inertia term.

k

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Wave frequency (WF) line tension (2)

• Morison force ( ) ( ) ( ) ( )d mf t k u t u t k u t 2

d u

m u

kk

k

Fatigue damage due to normalized Morison force (Madsen,1986)

10-2

10-1

100

101

102

0

0.5

1

1.5

2

2.5

3

k

0kGaussian

D

D

2

( ) ( ) ( )( )N

u u

u t u t u tf t k

Normalized:

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Amplitude distribution of LF and WF line tension

• The amplitude distribution of LF line tension shows a higher upper tail, which indicates a larger extreme value.

• While that of WF line tension is quite close to a Rayleigh distribution. Because in this case, the equivalent inertia term is dominating.

Scaled by the standard deviation of the fundamental process

( )max

X

X

pdf of the maximum of the variables

x0 1 2 3 4 5 6

f X(x

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Standard Rayleigh variableScaled maximum of LF mooring line tensionScaled maximum of WF mooring line tension

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Fatigue damage due to combined LF and WF line tension• The amplitude distribution of the process

with Gaussian and non-Gaussian cases

• The mean zero up-crossing rate can be obtained by the Rice formula.

( )P t

,0

(0) (0, )P P Ppf p dp

Scaled by the standard deviation of the fundamental process

( )max

X

X

( ) ( ) ( )LF WFP t X t R t

pdf of the maximum of the combined variables

x0 1 2 3 4 5 6 7 8

f X(x

)

0.0

0.1

0.2

0.3

0.4

0.5

0.6Standard Rayleigh+Rayleigh variableScaled maximum of LF+WF mooring line tension

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Short-term and long-term fatigue

LF WF Combined

RFC 0.040 0.855 1Non-Gaussian 0.059 0.854 1.030

Gaussian 0.046 0.921 1.080

• Short-term fatigue damage– A 3-hour sea state with

Hs=6.25m, Tp=12.5s, Uwind=7.5m/s, Ucurrent=0.5m/sConditional short-term fatigue damages

• Long-term fatigue damage– A smoothed northern North Sea

scatter diagram

29-year Joint PDF of Hs and Tp

Tp (sec)

0 5 10 15 20H

s (m

)0

2

4

6

8

100.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065

LF WF Combined

Non-Gaussian 0.065 0.812 1Gaussian 0.053 0.851 1.017

Total long-term fatigue damages

Smoothed scatter diagram(Joint density function)

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Long-term fatigue contribution

Non-Gaussian

Contour Graph 1

TP (s)0 5 10 15 20 25 30

HS (

m)

0

2

4

6

8

10

12

14

16

180.0000 0.0001 0.0002 0.0003 0.0004 0.0010 0.0012 0.0014 0.0016 0.0018

Contour Graph 1

TP (s)0 5 10 15 20 25 30

HS (

m)

0

2

4

6

8

10

12

14

16

180.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018

Contour Graph 1

TP (s)0 5 10 15 20 25 30

HS (

m)

0

2

4

6

8

10

12

14

16

180.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018

Contour Graph 1

TP (s)0 5 10 15 20 25 30

HS (

m)

0

2

4

6

8

10

12

14

16

180.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018

Contour Graph 1

TP (s)0 5 10 15 20 25 30

HS (

m)

0

2

4

6

8

10

12

14

16

180.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018

Contour Graph 1

TP (s)0 5 10 15 20 25 30

HS (

m)

0

2

4

6

8

10

12

14

16

180.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018

Gaussian

LF WF Combined

D=0.065 D=0.812 D=1

D=0.053 D=0.851 D=1.017

(sec)PT

( )SH m

(sec)PT

( )SH m

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Thank you !