Analytical Expressions of Capsize Boundary for a Ship with Roll Bias in Beam Waves

We have considered Melnikov's method for assessing ship safety against capsize in beam waves when there is some steady heel due, for example, to beam wind loading. For a generic restoring lever parameterized with respect to the degree of lever asymmetry ("bias") we have derived closed-form relationships characterizing capsizability and linking the critical wave-slope, the amount of bias, and the damping. Qualitative differences in the safe basin characteristics are identified, which depend on whether the bias is small or large. A critical look into the accuracy of the Melnikov formulas is also presented for nearly symmetric and for strongly asymmetric rolling, based on comparisons with numerical predictions.