It is well known that the full long-term response analysis is recognized as the most accurate and reliable method for evaluation of the extreme response in the design of ships and marine structures. However, such a method is time consuming for large and complex systems. To improve efficiency, the environmental contour method (ECM) is frequently used to approximate the long-term extreme response. The ECM is based on an environmental contour, which is traditionally obtained by the inverse first order reliability method (IFORM) with the assumption that the failure surface in the U space is approximated as a tangent plane at the design point. However, such an approximation underestimates the true failure probability if the failure surface in the U space is a concave set, and then the corresponding environmental contour would not be conservative for possible designs. In this work, a more conservative ISORM (inverse second order reliability method) contour is proposed. In this method, a specific secondorder surface is applied to approximate the failure surface at the design point, and then the failure domain in the U space would always be overestimated since the corresponding safe domain is approximated and underestimated as a sphere, regardless of the shapes of the failure surfaces. Therefore, the generated environmental contour can be always conservative for design purposes. The differences of the environmental contours generated by different methods, i.e., the traditional IFORM and the proposed ISORM, are illustrated by relevant examples, such as the wave statistics, wind wave statistics, and first-year ice ridge statistics.

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This article presents a four-dimensional (4D) path integration (PI) approach to study the stochastic roll response and reliability of a vessel in random beam seas. Specifically, a 4D Markov dynamic system is established by combing the single-degree-offreedom model used to represent the ship rolling behavior in random beam seas with a second-order linear filter used to approximate the stationary roll excitation moment. On the basis of the Markov property of the coupled 4D dynamic system, the response statistics of roll motion can be obtained by solving the Fokker-Planck equation of the dynamic system via the 4D PI method. The theoretical principle and numerical implementation of the current state of the art 4D PI method are presented. Moreover, the numerical robustness and accuracy of the 4D PI method are evaluated by comparing with the results obtained by the application of Monte Carlo simulation (MCS). The influence of the restoring terms and the damping terms on the stochastic roll response are investigated. Furthermore, based on the well-known Poisson assumption and the response statistics yielded by the 4D PI technique, evaluation of the reliability associated with high-level response is performed. The performance of the Poisson estimate for different levels of external excitations is evaluated by the versatile MCS technique.

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Currently development of floating wind turbines for deep water is mainly based on horizontal axis wind turbines (HAWTs). However, vertical axis wind turbines (VAWTs) are possible alternative due to their potential in the cost of energy reduction. This study deals with a comparison of stochastic dynamic responses of floating HAWTs and VAWTs with emphasis on the extreme structural responses and fatigue damages. A 5 MW three-bladed HAWT and three 5 MW VAWTs with blade number ranging from two to four were mounted on a semi-submersible platform. Their stochastic dynamic responses, short-term extreme structural responses and fatigue damages were estimated in both operational and parked conditions. The results show that the three-and four-bladed floating VAWTs and the three-bladed floating HAWTs considered have similar performances in the variation of generator power production, in the maximum tower base bending moment and in the fatigue damages at tower base and mooring lines. However, the maximum tensions in mooring line for the three-and four-bladed floating VAWTs are approximately four times higher than that of floating HAWTs, which implies a significant challenge for their mooring sys- *