Probability of Parametric Roll in Random Seaways

Abstract

The aim of the present chapter is to advocate for a very effective stochastic procedure based on the First-Order Reliability method (FORM) and Monte Carlo simulations (MCS), for prediction of parametric rolling in stationary stochastic seaways. Due to efficient optimization procedures implemented in standard FORM codes and the short duration of the time domain simulations needed, the calculation of the mean out-crossing rates of any given roll response is very fast. Thus complicated nonlinear effects can be included. Furthermore, the FORM analysis also identifies the most probable wave episodes leading to given roll responses. Because of the linearization in the FORM procedure the results are, however, only asymptotically exact and thus MCS often also needs to be performed. Here a scaling property inherent in the FORM procedure is investigated for use in MCS in order to reduce the necessary simulation time. More specifically, the MCS results for the reliability index β for parametric rolling of a ship suggest that the relation \(\beta = a + b/{H}_{\mathrm{s}}\) can provide an accurate scaling of the reliability index with significant wave height Hs. MCS can therefore be performed for sea states higher than the design sea state and thereafter be scaled down. From the reliability index the mean out-crossing rate and the probability of exceedance are then readily obtained.