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Fatigue under Bimodal Loads

Fatigue under Bimodal Loads. Zhen Gao Torgeir Moan Wenbo Huang. March 23, 2006. Contents. Bimodal random process Methods for bimodal fatigue damage assessment Case study of mooring system of a semi-submersible. Bimodal random process.

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Fatigue under Bimodal Loads

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  1. Fatigue under Bimodal Loads Zhen Gao Torgeir Moan Wenbo Huang March 23, 2006

  2. Contents • Bimodal random process • Methods for bimodal fatigue damage assessment • Case study of mooring system of a semi-submersible

  3. Bimodal random process • A wide-band process with a bimodal spectral density. • Examples: • Mooring line tension • Torque of propellers (or thrusters) in waves

  4. Fatigue based on S-N curve and Miner rule • Gaussian narrow-band fatigue damage • Fatigue damage of a bimodal process where is the mean zero up-crossing rate. is the standard deviation of the process.

  5. Methods for bimodal fatigue (1) • Level crossing counting • Rainflow counting • 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) • Dirlik (1985) • Benasciutti & Tovo (2003) • Jiao & Moan (1990) • Fu & Cebon (2000) • Huang & Moan (2006)

  6. Methods for bimodal fatigue (2) • Jiao & Moan (1990) • DNV (2005) Assume where is the envelope process of Then For Gaussian processes, analytical formula can be obtained.

  7. Methods for bimodal fatigue (3) • Fatigue damage estimation of a Gaussian bimodal process DNV (2005) Jiao & Moan (1990)

  8. Case study of mooring system • A semi-submersible Main particulars of the semi-submersible • Mooring system • Line No.10 • Pre-tension of 1320 kN • Studless chain link with a diameter of 125 mm Horizontal projection of the mooring system

  9. 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

  10. 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.

  11. 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) Basically, it is a Morison formula with a drag term and an equivalent inertia term. is a measure of the relative importance of the drag term and the equivalent inertia term.

  12. Wave frequency (WF) line tension (2) • Morison force Normalized: Fatigue damage due to normalized Morison force (Madsen,1986)

  13. Scaled by the standard deviation of the fundamental process ( ) 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.

  14. Scaled by the standard deviation of the fundamental process ( ) 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.

  15. Short-term and long-term fatigue • Short-term fatigue damage • A 3-hour sea state with Hs=6.25m, Tp=12.5s, Uwind=7.5m/s, Ucurrent=0.5m/s Conditional short-term fatigue damages • Long-term fatigue damage • A smoothed northern North Sea scatter diagram Total long-term fatigue damages Smoothed scatter diagram (Joint density function)

  16. Long-term fatigue contribution Combined WF LF D=0.065 D=0.812 D=1 Non-Gaussian D=0.053 D=0.851 D=1.017 Gaussian

  17. Thank you !

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