Probability of capsizing in beam seas with piecewise linear stochastic GZ curve

Abstract

The probability of capsizing for a dynamical system with time-varying piecewise linear stiffness is presented. The simplest case is considered, in which only the angle of the maximum of the restoring curve is changing. These changes are assumed to be dependent on wave excitation; such a system can be considered as a primitive model of a ship in beam seas, where all changes in stability are caused by heave motions. A split-time approach is used, in which capsizing is considered as a sequence of two random events: upcrossing through a certain threshold (non-rare problem) and capsizing after upcrossing (rare problem). To reflect the time-varying stability, a critical roll rate is introduced as a stochastic process defined at any instant of time. Capsizing is then associated with an upcrossing when the instantaneous roll rate exceeds the critical roll rate defined for the instant of upcrossing. A self-consistency check of the method, in which a statistical frequency of capsizing was obtained by time-domain evaluation of the response of the piecewise linear dynamical system and favorably compared with the theoretical prediction is described. © Springer Science+Business Media B.V. 2011.