Stochastic dynamic analysis and validation for the capsizing probability of a damaged vessel in random beam seas

Extreme roll motion response in realistic seas is a significant threat to ship stability, particularly for damaged vessels with weak maneuverability, as it increases the probability of capsizing. In this paper, a stochastic framework coupling nonlinear roll dynamics with flooding mechanics is established, providing new pathways for quantifying capsize probabilities in damaged vessel scenarios. A four-dimensional (4D) Markovian stochastic differential equation (SDE) which integrates a single degree of freedom (SDOF) roll motion equation with a linear filter model is constructed to study the nonlinear roll motion and the probability of a damaged vessel capsizing during progressive flooding state and steady state in random beam seas. A damage-flooding probability model is formulated through dynamic probabilistic coupling with the 4D SDE. The target vessel in this paper is DTMB 5415 which has symmetrical compartments. The accuracy of the 4D SDE solved by path integration (PI) method is validated by comparing the exceedance rates with those obtained from direct counting method based on 5 DOF time-domain numerical model for damaged vessels. The conclusion is that the exceedance rates obtained from 4D SDE have consistent trend and are more conservative compared to the results from direct counting method. • A stochastic differential equation (SDE) is proposed for capsizing probability calculation of damaged vessels. • The hydrostatic and hydrodynamic moments in SDE are influenced by flooding water mass. • The path integration method is employed to effectively solve the Fokker-Planck equation associated with the SDE. • The exceedance rates calculated through path integration method are compared with those calculated by direct counting method.