Applied Ocean Research 29 (2007) 45–54 www.elsevier.com/locate/apor Fatigue damage induced by nonGaussian bimodal wave loading in mooring lines Zhen Gao * , Torgeir Moan Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, Otto Nielsens v 10, N-7491, Trondheim, Norway Received 20 October 2006; received in revised form 14 May 2007; accepted 1 June 2007 Available online 23 July 2007 Abstract Catenary mooring lines are typically subjected to bimodal loads, comprising of a wave frequency (WF) component due to the first-order wave forces and a low frequency (LF) component induced by the second-order wave forces. For moored vessels, the LF forces due to current and wind also play a role. Only dynamic wave loads are considered herein, while current and wind loads are modeled as constant forces. Because of the nonlinearities of the mooring line characteristics, the dynamic line tension and the second-order responses, both the WF and LF line tensions are in principle nonGaussian. These facts make it difficult to estimate the combined fatigue damage of mooring lines in the frequency domain. A fatigue combination rule based on the Jiao and Moan’s theory has been extended to cover the nonGaussian case. The purpose of this paper is to improve and validate the frequency-domain method by time-domain analysis based on a simplified, but accurate mechanical model of the dynamic line tension. Improvements on the LF and combined fatigue damage estimation have been made by considering the nonsymmetrical property of the LF line tension distribution. Both the WF and LF mooring line tensions due to wave loading have been simulated in the time domain for different sea states and the combined fatigue damage has been estimated by using the rainflow cycle counting algorithm. The accuracy of the frequency-domain method for estimating the bimodal nonGaussian fatigue damage of mooring lines has been verified by the time-domain simulations and is considered to be acceptable. c  2007 Elsevier Ltd. All rights reserved. Keywords: Fatigue damage; NonGaussian loading; Bimodal process; Frequency domain; Time domain; Rainflow cycle counting 1. Introduction Catenary mooring lines are pre-tensioned and subjected to dynamic loads caused by the wave frequency (WF) vessel motions and the low frequency (LF) motions normally occurring in the horizontal plane. The WF vessel motions in all six degrees of freedom are Gaussian and can be very well predicted by using linear transfer functions without accounting for the effect of mooring system. However, the LF vessel motions induced by waves are typically nonGaussian due to the nonlinear second-order slowly-varying wave forces (as well as the interactions with LF varying wind and current) and are coupled with the mooring line dynamics. Dynamic mooring line tension depends on the vessel motions and the line characteristics. Usually, as the same as the tension due to the mean forces of waves (mean wave drift), wind * Corresponding author. Tel.: +47 73595736; fax: +47 73595528. E-mail addresses: zhen.gao@marin.ntnu.no (Z. Gao), torgeir.moan@marin.ntnu.no (T. Moan). and current, the LF line tension can be quasistatically estimated because of the relative large oscillation periods. Due to the narrow-band property of the LF motion response, the LF line tension is also narrow-banded. In order to accurately calculate the tension induced by the WF vessel motion, a dynamic model of mooring line must be applied, e.g. the simplified model proposed by Larsen and Sandvik [1]. Due to the drag force acting on the line and the geometrical nonlinearity, the WF line tension is in principle nonGaussian as well. Moreover, the WF line tension typically has a dominating central frequency equal to that of the WF vessel motion. Therefore, the WF line tension could also be assumed to be narrow-banded. As a consequence, the combined WF and LF tension is a nonGaussian wide-band process, although the individual components can both well be assumed to be narrow-banded. An appropriate estimation of the effective stress range distribution of the total process is crucial for fatigue analysis under wide-band loads. For bimodal Gaussian processes, several frequency-domain combination rules are available and Benasciutti and Tovo [2] discussed and compared these 0141-1187/$ - see front matter c  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.apor.2007.06.001