Calculation of capsizing probability for damaged vessels in beam seas based on direct counting method and dynamic model

This study aims to calculate and analyze the capsizing probability of damaged vessels in beam seas under dead ship conditions, considering varying significant wave heights and characteristic wave periods. The analysis is conducted using the direct counting method and a dynamic model. The objective is to refine and improve the existing conservative estimation method based on residual stability parameters used in damaged vessel stability assessment regulations.

The dynamic model incorporates the coupled interactions among waves, the vessel, and internal floodwater. The wave-ship coupling is modeled by calculating nonlinear restoring forces and Froude−Krylov forces through instantaneous integration of pressures over the wetted surface. Radiation and diffraction forces are obtained using the STF method in combination with impulse response functions, while experimentally derived roll damping coefficients are included to capture viscous effects. The wave-floodwater coupling is accounted for through modified Bernoulli equations that determine inflow and outflow rates at damaged openings. The ship-floodwater coupling is modeled using the lumped mass method to simulate floodwater sloshing forces. Hydrostatic pressures induced by flooding are evaluated via instantaneous integration over the wetted surface, and viscous dissipation effects are approximated using semi-empirical formulations. The direct counting method, following the International Maritime Organization (IMO) guidelines, is employed to quantify capsizing probability using time-dependent marginal probability functions.

The proposed model is applied to the benchmark damaged vessel DTMB 5415 to evaluate roll motion responses and capsizing probabilities. The key findings are as follows: 1) The dynamic model’s predicted roll motion responses exhibit relative errors within 20% compared to experimental data near the period corresponding to the peak response. 2) The 30-minute capsizing probability increases with significant wave height, reaching its maximum when the wave characteristic period approaches the natural roll period of the damaged vessel.

The proposed methodology provides an objective and accurate approach for predicting the capsizing probability of damaged vessels in beam seas. It demonstrates significant improvements over conventional residual stability-based approaches.