Estimation of hull girder vertical bending moments including non-linear and flexibility effects using closed form expressions

A simple but rational procedure for prediction of extreme wave-induced hull girder bending moment in slender mono-hull displacement vessels is presented. The procedure takes into account main ship hull characteristics such as length, breadth, draught, block coefficient, bow flare coefficient, forward speed, and hull flexibility. The wave-induced loads are evaluated for specific operational profiles. Non-linearity in the wave bending moment is modelled using results derived from a second-order strip theory and water entry solutions for wedge-type sections. Hence, bow flare slamming is accounted for through a momentum type of approach. The stochastic properties of this non-linear response are calculated through a monotonic Hermite transformation. In addition, the impulse loading attributable to, for example, bottom slamming or a rapid change in bow flare is included using a modal expansion in the two lowest vertical vibration modes. These whipping vibrations are added to the wave frequency non-linear response, taking into account the rise time of the impulse response as well as the phase lag between the occurrence of the maximum non-linear load and the maximum impulse load. Previous results for the sagging bending moment are validated by comparison with fully non-linear strip theory calculations and supplemented with new closed form results for the hogging bending moment. Focus is on the extreme hull girder hogging bending moment. Owing to the few input parameters, this procedure can be used to estimate the wave-induced bending moments at the conceptual design phase. Another application area is for novel single-hull ship types not presently covered by the rules of the classification societies. As one application example the container ship MSC Napoli is considered. Further validations are needed, however, in order to select proper values of the parameters entering the analytical form of the slamming impulse.

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