## 6 Conclusions  
In this paper, a time-domain hydroelastic method for wave-induced ship nonlinear motions and loads prediction is presented. The hull girder is modeled as a 1D flexible beam so that the hydroelastic responses and slamming induced whipping responses are properly considered. Then a nonlinear long-term statistics method in terms of the probability of exceedance and extreme value is presented based on the time-domain nonlinear short-term simulation results. The calculated long-term vertical sagging and hogging moments amidships are further used to provide classification society Rules for ultimate strength assessment. The following conclusions can be obtained from this study:  
1. The developed time-domain hydroelastic theory, which combines 3D BEM and 1D beam model is reliable in the prediction of ship nonlinear motion and load responses and it has been well  
Figure 23: The calculated curve of VBM versus curvature. \begin{table}
\begin{tabular}{c c c c} \hline
**Item** & **Symbol** & **Hogging** & **Saging** \\ \hline Calm water VBM & \(M_{S}\) (M\(\mathrm{N}\mathrm{m}\mathrm{m}\)) & 3688 & 2948 \\ Wave-induced VBM & \(M_{\mathrm{W}}\) (M\(\mathrm{N}\mathrm{m}\)) & 4627 & 5366 \\ Extreme VBM by nonlinear direct calculation & \(M_{\mathrm{w}\mathrm{i}\mathrm{p}}\) (M\(\mathrm{N}\mathrm{m}\)) & 9036 & 10,382 \\ Extreme VBM by linear direct calculation & \(M_{\mathrm{w}\mathrm{i}\mathrm{e}\mathrm{r}}\) (M\(\mathrm{N}\mathrm{m}\)) & 6971 & 7533 \\ Correction factor due to whipping loads & \(f_{\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{p}}\) & 1.30 & 1.38 \\ Safety factor of \(M_{S}\) & \(\gamma_{S}\) & 1.00 & 1.00 \\ Safety factor of \(M_{W}\) & \(\gamma_{W}\) & 1.10 & 1.10 \\ Safety factor of \(M_{U}\) & \(\gamma_{R}\) & 1.15 & 1.15 \\ Safety factor of material uncertainty & \(\gamma_{M}\) & 1.02 & 1.02 \\ Safety factor of condition simplification & \(C_{\mathrm{m}\mathrm{i}\mathrm{e}\mathrm{r}}\) & 1.01 & 1.01 \\ Ultimate bearing capacity VBM & \(M_{U}\) (M\(\mathrm{N}\mathrm{m}\)) & 14,713 & 14,550 \\  & \(M_{\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{d}}\) (M\(\mathrm{N}\mathrm{m}\)) & 10,305 & 11,094 \\ Strength check by Rule approach & Failure ratio & 82\% & 88\% \\  & Safety satisfy? & Yes & Yes \\  & \(M_{\mathrm{T}\mathrm{e}\mathrm{i}\mathrm{d}}\) (M\(\mathrm{N}\mathrm{m}\)) & 9036 & 10,382 \\ Strength check by direct calculation & Failure ratio & 72\% & 83\% \\  & Safety satisfy? & Yes & Yes \\ \hline \end{tabular}
\end{table} 
Table 5: Hull girder ultimate strength assessment results. validated by comparison with the tank experimental results. Since the developed hydroelastic code runs much faster than the more complicated algorithm, which combines 3D BEM and 3D FEM, it is applicable for direct calculation of extreme wave loads and other nonlinear responses by performing long time numerical simulation. 2. The slamming induced whipping loads acting on ship when sailing in harsh waves and at high speed significantly increase the extreme loads for head or bow oblique wave conditions, while the whipping loads are very weak when ship sailing in beam or stern oblique waves. The long-term extreme VBM load predicted by nonlinear hydroelastic theory considering slamming effects is approximately 30% higher than that by linear hydroelastic theory. The simplification of simulation condition has ignorable small influence on the long-term extreme values. 3. Due to the fact that the effects of whipping loads on a hull girder's ultimate strength is not well considered in the majority of the present Rule Book approaches, a simplified strength check formula, which includes the whipping load coefficient, is developed on the basis of the design wave loads by classification society Rule approach. The developed strength check formula is very useful for the hull girder ultimate strength assessment of a large flexible ship with a flare bow operating in harsh conditions.