1 – Centre for Ships and Ocean Structures 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 Gao Prof. Torgeir Moan Centre for Ships and Ocean Structures, Norwegian University of Science and Technology February 24, CeSOS – Centre for Ships and Ocean Structures

2 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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) Bimodal (left) and trimodal (right) spectra Benasciutti and Tovo (2005)Transfer function (left) and wave spectrum (right)

5 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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 10% over-estimation Maximum 30% over-estimation Larger variation Accuracy of the narrow-band fatigue approximation (2)

6 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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) Accuracy of the freq.-d. method for fatigue analysis of wave-induced responses Wave spectrum (up); Transfer function (down)

7 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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 is the envelope process of Define

8 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures Bimodal fatigue analysis (2) Comparison with the rainflow counting method w1w1 w2w2 var 1 var 2 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 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures Multi-modal fatigue formulation (3) Comparison with the rainflow counting method var 1 var 2 var 3 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 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures var 1 w1w1 var 2 var 3 w2w2 w3w3 var 1 =var 2 =var 3 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 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures Multi-modal fatigue formulation (5) Spectral density function (Benasciutti & Tovo, 2005) Accuracy of the freq.-d. method for general wide-band fatigue analysis Case study of generally defined wide-band spectra

14 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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 (2005) - API RP 2SK (2005) - DNV OS-E301 (2004) Wind Wave Current Original Position Mean Position Dynamic Analysis (WF+LF) Static Analysis

15 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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 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)

16 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures WF mooring line tension (Morison formula) Amplitude distribution (Borgman, 1965) Application of non-Gaussian bimodal fatigue analysis to mooring line tension (3) Amplitude distribution of WF tension where Drag dominant (Exponential) Inertia dominant (Rayleigh) where Combined Rayleigh and exponential distribution!

17 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures Application of non-Gaussian bimodal fatigue analysis to mooring line tension (4) Distributions of the fundamental variables 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 Skewness>0 Kurtosis>3

18 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures Comparison of the frequency-domain method for fatigue analysis with time-domain simulations (Gao & Moan, 2007) –Short-term sea states: Hs= m Tp= sec –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 Application of non-Gaussian bimodal fatigue analysis to mooring line tension (5)

19 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures 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

20 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures Application in design code –Mooring system (ISO , 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 Spectra of overturning moment An example of multi-modal response of offshore fixed wind turbines

21 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures [1] Jiao, G. & Moan, T. (1990) Probabilistic analysis of fatigue due to Gaussian load processes. Probabilistic Engineering Mechanics; Vol. 5, No. 2, pp [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 [3] Fu, T.T. & Cebon, D. (2000) Predicting fatigue lives for bi-modal stress spectral densities. International Journal of Fatigue; Vol. 22, pp [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. OMAE [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. OMAE [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 , pp [7] Olagnon, M. & Guede, Z. (2008) Rainflow fatigue analysis for loads with multimodal power spectral densities. Marine Structures; Vol. 21, pp [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 [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 References (1)

22 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures [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 [12] Bouyssy, V., Naboishikov, S.M. & Rackwitz, R. (1993) Comparison of analytical counting methods for Gaussian processes. Structural Safety; Vol. 12, pp [13] Benasciutti, D. & Tovo, R. (2005) Spectral methods for lifetime prediction under wide-band stationary random processes. International Journal of Fatigue; Vol. 27, pp [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 [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 [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 [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 [20] Næss, A. (1986) The statistical distribution of second-order slowly-varying forces and motions. Applied Ocean Research; Vol. 8, No. 2, pp [21] Gao, Z. & Moan, T. (2007) Fatigue damage induced by non-Gaussian bimodal wave loading in mooring lines. Applied Ocean Research; Vol.29, pp References (2)

23 – Centre for Ships and Ocean Structures Gao & Moan – Centre for Ships and Ocean Structures Thank you!