Dr. Zhen Gao Prof. Torgeir Moan

1. Frequency-domain multi-modal formulation for fatigue analysis of Gaussian and non-Gaussian wide-band processes Dr. Zhen Gao Prof. 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. Contents 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

3. Introduction • 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)

4. Accuracy of the narrow-band fatigue approximation (1) Bimodal (left) and trimodal (right) spectra 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) Transfer function (left) and wave spectrum (right) Benasciutti and Tovo (2005)

5. Accuracy of the narrow-band fatigue approximation (2) • 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) The NB approximation is too conservative for these spectra! Maximum 30% over-estimation Larger variation Maximum 10% over-estimation

6. Accuracy of the narrow-band fatigue approximation (3) • 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 freq.-d. method for fatigue analysis of wave-induced responses 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 the mean zero up-crossing rate the amplitude distribution Define is the envelope process of

8. var1 var2 w1 w2 Bimodal fatigue analysis (2) • Comparison with the rainflow counting method • 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

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

11. Multi-modal fatigue formulation (3) • Comparison with the rainflow counting method var1 var3 var2 • 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 • Accuracy of the freq.-d. method for trimodal fatigue analysis

12. var2 var3 var1 var1=var2=var3 w3 w1 w2 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) • Case study of generally defined wide-band spectra Spectral density function (Benasciutti & Tovo, 2005) • Accuracy of the freq.-d. method for general wide-band fatigue analysis

14. Dynamic Analysis (WF+LF) Wind Original Position Mean Position Wave Static Analysis Current 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)

15. Application of non-Gaussian bimodal fatigue analysis to mooring line tension (2) • 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

16. Application of non-Gaussian bimodal fatigue analysis to mooring line tension (3) where Drag dominant (Exponential) Inertia dominant (Rayleigh) where Combined Rayleigh and exponential distribution! Amplitude distribution of WF tension • WF mooring line tension (Morison formula) • Amplitude distribution (Borgman, 1965)

17. Application of non-Gaussian bimodal fatigue analysis to mooring line tension (4) • 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) LF line tension Second-order wave forces Linearized model LF vessel motions Offset-tension curve Skewness>0 Kurtosis>3 Distributions of the fundamental variables

18. Application of non-Gaussian bimodal fatigue analysis to mooring line tension (5) • 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% Black – WF; Red – LF;Green – Combined fatigue Accuracy of the freq.-d. method for fatigue analysis

19. Conclusions 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.

20. Recommendations for future work An example of multi-modal response of offshore fixed wind turbines • 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 Spectra of overturning moment

21. References (1) [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.

22. References (2) [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.

23. Thank you!