tical moments only (mean, variance, skew and kurtosis).

The similar assumption was made by Winterstein [13]. Using Hermite moment formulations approximate probability density functions, crossing rates and extreme values were derived solely based on the aforementioned statistical moments.

In summary, it was shown by different researchers that the statistics of weakly nonlinear stationary seakeeping problems are reasonably well defined by the first four statistical moments only. This paper shows the applicability of an approximate third-order Volterra modelling to analyze the statistics of the vertical hull girder loads in irregular waves. The numerical results are compared with extensive model tests in irregular waves. The comparisons comprise time traces, power spectra and spectral moments, statistical moments and probability density functions of the samples and of the peak-peak values. The required linear, quadratic and cubic frequency response functions were derived from the first three harmonic components measured in regular waves.

The first results of systematical experiments, focussed on the nonlinear vertical hull girder loads, were presented by Dalzell [14,15] in 1964. Models of three variants of a Mariner ship, a tanker and a destroyer were subjected to a range of regular waves over a range of wave lengths and heights. The vertical bending moments were presented in hogging and sagging condition separately, not providing information about the harmonic components in the response signals. However, it was proved without a doubt that the sag/hog-ratio was not equal to unity, which should be the case for linear signals. Furthermore the experiments showed that the sag/hog-ratios tended to be larger for the slender destroyer model and the Mariner variants than for the full tanker model. Similar conclusions followed from two other model test series, reported by Murdey [16] and Nethercote [17].

O'Dea et al [18] reported the measurement of nonlinear heave and pitch responses for a S-175 model. The higher harmonic components were only a few percent in magnitude of the first harmonic response. This seems to be a negligible effect. It has to be realised, however, that the accelerations are more strongly nonlinear than the displacements when we compare them with the magnitude of their linear components. This can be illustrated on the assumption of a third order, zero mean periodic displacement, which is written in terms of the first three harmonic components as

(1)

Hence, the displacement, velocity and acceleration are given in matrix notation by

(2)

It can easily be seen that relative to the first harmonic component, the second harmonic acceleration is four times as large as the second harmonic displacement while the third harmonic component is even nine times larger. This much more pronounced nonlinear inertia effect directly influences the hull girder loads behaviour.

After a survey of literature it had to be concluded that the data sets presented were not sufficient to study the nonlinear hull girder loads in very much detail. Many of the experiments were performed in regular waves only. From those test results, too much information was lost due to the presentation of the results in terms of hog/sag-ratios or double amplitudes. No systematical results were presented showing the harmonic components of a response experienced in regular wave conditions in order to investigate the actual order of the process.

Therefore new extensive experiments were performed both in regular and irregular waves. The results were extensively reported and discussed by Adegeest [19,20,21]. The objective of the experiments was to collect motion and load data that can be studied and compared with numerical solutions in much more detail.

The experiments were conducted on a Wigley hull form with and without bow flare. The normalised beam *y* of the under water ship is described by a polynomial in the *x*- and *z*-coordinate according to

*y*=(1–*z*^{2})(1–*x*^{2})(1+0.2*x*^{2}) +*z*^{2}(1–*z*^{8})(1–*x*^{2})^{4} (3)

where *x* ∈ [–1,1] and *z* ∈ [–1,0]. Table 1 shows the main characteristics of the Wigley geometries. The bow form variation is clearly illustrated in figure 1.

In regular waves both models were tested at two forward speeds, thirteen frequencies and at least four wave amplitudes. Fourier analyses of the results clearly showed the presence of pronounced higher harmonic components in the hull girder load