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A typical parametric roll scenario for a ship in head waves implies that the roll motion is coupled with vertical motion of the vessel. The added resistance of the ship is increased when the bow pitches down in a wave crest. As a consequence, the ship speed is slowed down and, hence, the roll resonance condition might be changed. In an attempt to study the influence of this speed variation in waves on parametric roll, the procedure for estimation of probability of parametric roll by Jensen and Pedersen (2006) has been extended to account for the surge motion of the vessel.

This paper analyses the wave-induced high-frequency bending moment response of ships, denoted springing. The aim is to predict measured severe springing responses in a large bulk carrier. It is shown that the most important springing contribution is due to the resultant second order excitation in multidirectional sea. The incident pressure field from the second order bidirectional wave field is derived, including the non-linear cross-coupling terms between the two wave systems (e.g. wind driven waves and swell). The resulting effect of the super-harmonic cross-coupling interaction terms on the springing response is discussed. An example with opposing waves is given, representing probably the 'worst' case for energy exchange between the wave systems. Theoretical predictions of standard deviation of wave- and springing-induced stress amidships are compared with full-scale measurements for a bulk carrier.

The present paper considers a large ocean going ship with significant springing responses, which have made a large contribution to the fatigue cracking for certain structural details. Four different theories for predicting ship responses and associated computer programs for predictions of springing are described. These theories represent four different approaches with various characteristics, e.g. linear, scond-order, nonlinear, frequency-domain, time-domain, two-dimensional and three-dimensional, in calculating hydrodynamic loads and vibrations. The numerical programs, WASIM (DNV), SOST (DTU), SINO (CSSRC) and VERES (Marintek), have been well validated for ordinary ship responses. Assumptions regarding how the different programs are sued in the present calculations are provided. Sensitivity studies are carried out and the main results are presented. A selected number of full-scale measurements with various sea states and headings used for comparison between numerical results and the measurements are described. The comparison bwteeen simulated and measured high frequency (2-node vibration, denoted HF) and low frequency (wave-induced, denoted LF) stresses in addition to some important observations are made.

The wave-induced bending moment in ships is the most important sea load parameter for ships larger than 100m in length. Hence, any rational ship design procedure must include a reasonable accurate determination of this load and a large amount of various hydrodynamic formulations have been published, ranging from semi-empirical formulas to three-dimensional non-linear procedures. A review of the state-of-the art can be found in ISSC.VI.1 (2000). These procedures must be combined with operational and sea state information to predict the probability distribution of the maximum wave-induced bending moment a ship may be subjected to during its operational lifetime. Whereas the influence of forward speed and ship heading with respect to the waves usually is accounted for, the effect of water depth is seldom considered, except in non-linear time domain formulations where a confined water domain must be specified anyhow. Usually, two-dimensional strip theories, either linear or non-linear, are applied for actual design cases and these theories are normally based on incident deep-water waves and furthermore apply added mass and damping calculations based on infinite water depth. Only a few papers have in the past addressed the influence of water depth on the ship response. In an early work Kim (1968) presented results for the variation of the added mass and hydrodynamic damping and for the heave and pitch motion for a Series 60 model using a relative motion strip theory formulation. A significant reduction in ship motion with decreasing keel clearance was observed. In the present paper a rigorous implementation of finite water depth in the consistent linear strip theory by Salvesen et al. (1970) is presented together with results for the variation of the motion and wave-induced bending moment with water depth for a container vessel. The results show that if the water depth is less than two times the draft of the vessel, the wave-induced bending moment becomes significant lar