## 4 Statistical Analyses of Numerical Simulation Results  
### Calculation Method and Procedure  
The time series of short-term motion and load responses under specified wave state, navigational speed, wave heading and loading condition can be calculated using the developed time-domain hydroelastic code. The crest and trough peak values in each zero-up-crossing period are counted and then they are used to derive the statistical values such as single significant amplitude or root mean square. Moreover, the Weibull distribution function is used to fit the probability density distribution of the peaks obtained in each short-term period.  
An example of the short-term statistical analysis using the numerical data of VBM amidships in Condition 2 is shown as follows. Figure 13 presents the simulated time series of VBM within 1 h and the 100 s local view from 1200 s to 1300 s with explanation of the statistical values. It is noted that the short-term predictions are conducted respectively by adopting the nonlinear hydroelasticity theory considering slamming loads and the linear hydroelasticity theory without considering slamming loads. Figure 14 presents the corresponding histogram of the statistical hogging and sagging VBM peak values as well as the Weibull fitting results. In addition, Figures 15 and 16 present the histogram of the statistical crest and trough values of heave and pitch motions as well as the Weibull fitting results. It is noted that the presented motion results are obtained by nonlinear hydroelastic numerical simulation and the motion signals are turned out to be associated with less nonlinearity compared with the loads. The distributions of heave and pitch peaks are close to Rayleigh distribution, which is a kind of special Weibull distribution.  
The long-term motion and load responses under a series of different wave states, navigational speeds, wave headings and loading conditions can be predicted based on the short-term prediction results according to the method described in Section 2.3. Moreover, in order to identify the influence of whipping loads on hull girder ultimate strength, the long-term extreme loads that obtained by nonlinear hydroelasticity theory and linear hydroelasticity theory are compared and the ratio coefficient of extreme loads is used to describe the influence of whipping loads on total loads. The ratio coefficient is also used for subsequent hull girder ultimate strength assessment in combination with classification society Rules approach. The flowchart of the long-term prediction of extreme loads and subsequent structural ultimate strength assessment is summarized in Figure 17.  
Figure 13: Statistics of peak values from the short-term simulation results. (**a**) The simulated time series in 1 h; (**b**) zoom view of time series in 100 s.  
Figure 16: The histogram of pitch peak distribution. (**a**) Nonlinear pitch crest; (**b**) nonlinear pitch trough.  
Figure 14: The histogram of VBM peak distribution. (**a**) Nonlinear hogging VBM; (**b**) nonlinear sagging VBM; (**c**) linear hogging VBM; (**d**) linear sagging VBM.  
Figure 15: The histogram of heave peak distribution. (**a**) Nonlinear heave crest; (**b**) nonlinear heave trough.