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In this article, a mathematical model based on the 2D+T theory has been developed to predict the performance of two-stepped planing hulls in calm water. It has been attempted to develop a mathematical model without using regression formulas. It leads to development of a computational model with no common limitations related to empirical models which have an individual range of applicability. For this purpose, theoretical solution of water entry of a two-dimensional wedge section has been implemented to compute the pressure distribution over wedge section entering water, and then normal forces acting on the two-dimensional sections are computed. Bottom of the boat has been divided into three different planing surfaces including fore, middle and aft bodies. Computations are performed for each of these surfaces. By integrating the two-dimensional sectional normal forces over the entire wetted length of the vessel, the trim angle, wetted surface and resistance have been obtained. To evaluate the accuracy of the presented method, the obtained results are compared against experimental data and a previous empirical-based method developed by authors. The comparison suggests that the proposed method predicted dynamic trim angle, wetted surface and resistance of double stepped boats with reasonable accuracy. The mean errors in prediction of trim angle, wetted surface and resistance are, respectively, 13%, 16% and 8%. It should also be noted that although computation of running attitudes and resistance of double-stepped planing boats are targeted in this article, the mathematical model has been developed in such a way that it has the potential to model transverse and vertical motions of two-stepped planing hulls in future studies.

In the current article, the hydrodynamic forces of single-stepped planing hulls were evaluated by an analytical method and compared against towing tank tests. Using the 2D + T theory, the pressure distribution over the wedge section entering the water and the normal forces acting on the 2D sections have been computed. By integrating the 2D sectional normal forces over the entire wetted length of the vessel, the lift force acting on it has been obtained. Using lift forces as well as the consequence pitch moment, the equilibrium condition for the single-stepped planing hull is found and then resistance, dynamic trim, and the wetted surface are computed. The obtained hydrodynamic results have been compared against the experimental data and it has been observed that the presented mathematical model has reasonable accuracy, in particular, up to Froude number 2.0. Furthermore, this mathematical model can be a useful and fast tool for the stepped hull designers in the early design stage in order to compare the different hull configurations. It should also be noted that the mathematical model has been developed in such a way that it has the potential to model the sweep-back step and transverse the vertical motions of single-stepped planing hulls in future studies.

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