Including Roll in the Evaluation of Bending Loads for the Hull Girder in a Seaway

The longitudinal strength of the hull girder is traditionally verified by linear superposition of still water (static) loads to wave induced (dynamic) loads. Belonging to the former class, gravitational effects are considered always as directed along the plane of symmetry of the ship, as the static equilibrium position corresponds to the ship floating with a neutral heel angle. This is not the case for a ship in a seaway, as the weight will act in reality along the instantaneous ‘true vertical’, which is not in general contained in the symmetry plane, due to ship motions. The actual direction of the weight force represents one of the non-linear aspects of the seakeeping problem, as the direction of one of the input external forces (weight) depends on the instantaneous roll angle (representing an output of the study). The present paper investigates the above subject, trying to quantify this retro-action effect with reference to the verification of the bending strength of the hull girder. The attention is focussed on the magnitude of the horizontal bending component generated by the inclined direction of the weight force as compared to the bending component induced by waves in the same plane (normal to the plane of symmetry of the ship). The subject is discussed on the basis of a linear seakeeping analysis, which is not able to model properly the non-linear aspects above recalled, but provides anyway as output an estimate of roll motions that can be used to evaluate in a qualitative way how large would be the horizontal load component due to weight, not accounted for in equilibrium equations. The statistical correlation between the wave induced bending load and roll angle is treated with the simplified assumption of a Gaussian joint distribution and a consistent model for the combination of horizontal and vertical bending components is established. The reference situation is chosen as corresponding to the extreme value of the vertical wave induced component, and distribution for the horizontal wave bending component and of roll motion are conditioned to the occurrence of such extreme value. This can be seen as an extension of Turkstra’s rule for the combination of loads to the case of correlated loads. The stochastic model is applied in the evaluation of the hull girder reliability for a test case and the conditioned probability distribution of roll motion at the failure point is derived in order to discuss the influence of a possible inclusion of roll effects in the verification of the hull girder bending strength.