Study on simplified estimation formula for long-term predictions of ship response in waves – Application to vertical bending moment

This paper proposed a simplified formula for the long-term maximum wave-induced vertical bending moment (VBM) of ships. First, as a general theory independent of the type of ship response, the relationship between the long-term response and the peak value of the response amplitude operation (RAO) was discussed and its governing factors were clarified, and the methodology of the simplified formula development approach was presented. In particular, if the RAO has two properties, “band parameter-invariance” and “peak value-wavelength correspondence,” the long-term response can be expressed by only the peak value of RAO. Based on this approach, a simplified formula for the long-term maximum VBM was presented. First, focusing on the characteristics of the VBM-RAO in the case of zero speed, a formula with zero speed was derived, and based on this, a formula with a speed of 5 knots was derived. These formulae were validated by comparison with the results of a numerical analysis using 154 actual merchant ship models. The proposed method for developing the simplified formula is physically clear and is valid for all types of ships and responses. •A simplified formula for the linear long-term maximum response of ships in a general form, regardless of the ship type and size, is presented.•The long-term response corresponds to the peak value of the response amplitude operator (RAO).•The formula is expressed very simply when the non-dimensional RAO has two characteristics: band parameter-invariance and peak value-wavelength correspondence.•Simple and accurate formulae of long-term vertical bending moment were derived for the 0 knot and 5 knots cases.