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

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.

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)

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.

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

Case study of mooring system A semi-submersible Main 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 diameter of 125 mm Horizontal projection of the mooring system

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

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.

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.

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

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 ( )

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

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 LF WF Combined RFC 0.040 0.855 1 Non-Gaussian 0.059 0.854 1.030 Gaussian 0.046 0.921 1.080 Long-term fatigue damage A smoothed northern North Sea scatter diagram Total long-term fatigue damages LF WF Combined Non-Gaussian 0.065 0.812 1 Gaussian 0.053 0.851 1.017 Smoothed scatter diagram (Joint density function)

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

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