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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 paper presents a simple method for predicting snow density by use of weather data. Six hundred and eight snow density (bulk weight density) measurements from the period 1967-1986 are used in a multiple regression analysis. The measurements are performed at 105 sites in theÅs area situated in Akershus County in southeast Norway. The area has a relatively stable winter climate. Weather data from an observing station with three registrations a day are used in the analyses. The distance between the measurement sites and the meteorological station varied between 0.6 and 14 km. A clear correlation is found between the observed climate and the measured snow density. A multiple regression equation is developed with a coefficient of determination equal to 70% and a standard deviation equal to 24 kgm −3 . Snow density values suggested in the Norwegian code NS 3491-3 is in most cases overestimated. Expressions given in an annex to the international regulations for snow loading on roofs (ISO 4355) are less applicable to prescribing snow density for climate studied in this investigation. Still, they can be used as simple and rough estimates.

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.