## 2 Ship Hydroelasticity Theory in Irregular Waves
### Prediction and Statistics of Extreme Values  
The long-term statistics of wave-induced motions and loads is well established in linear frequency-domain theory. The short-term response of ship linear motion and load is a stationary Gaussian narrow-band process with zero mean, and therefore their peaks obey Rayleigh distribution. The long-term distribution is then obtained by superposition of short-term probabilities of exceedance in all possible combinations of wave states, heading angles, navigational speeds and loading conditions.  
However, the nonlinear motion and load response is no longer Gaussian and the peak distribution cannot be described by Rayleigh distribution. Moreover, the hogging and sagging VBM reveal obvious asymmetry due to the effects of fluid dynamics and slamming loads. Therefore, the short-term statistics should be processed using time series analysis method rather than the spectral analysis method. The time series analysis method is described in Section 4.1 and the following statistical method is used to obtain the probability distribution of the counted peaks. Suppose the random variable \(X\) is of Weibull distribution, then the probability density function and cumulative distribution function of variable \(x\), which represents the statistical peak values (e.g., crest or trough), can be, respectively, expressed as follows:  
\[f_{0}(x)=\frac{l}{k}\Big{(}\frac{x}{k}\Big{)}^{l-1}e^{-(\frac{x}{k})^{l}} \tag{18}\]  
\[F_{0}(x)=1-e^{-(\frac{x}{k})^{l}} \tag{19}\]  
where \(l\) and \(k\) are the two parameters of Weibull distribution.  
Suppose that the loading condition of ship does not change, the probability density function and cumulative distribution function for long-term prediction can be obtained by summation of short-term results in all possible combinations of the independent wave states, heading angles and navigational speeds:  
\[f(x)=\frac{\sum\limits_{i}\sum\limits_{j}\sum\limits_{k}n_{0}^{ijk}p_{i}(H_{1/3},T_{z})p_{j}(\beta)p_{k}(U)f_{0}^{ijk}(x)}{\sum\limits_{i}\sum\limits_{j}\sum \limits_{k}n_{0}^{ijk}p_{i}(H_{1/3},T_{z})p_{j}(\beta)p_{k}(U)} \tag{20}\]  
\[F(x)=\frac{\sum\limits_{i}\sum\limits_{j}\sum\limits_{k}n_{0}^{ijk}p_{i}(H_{1/3 },T_{z})p_{j}(\beta)p_{k}(U)F_{0}^{ijk}(x)}{\sum\limits_{i}\sum\limits_{j}\sum \limits_{k}n_{0}^{ijk}p_{i}(H_{1/3},T_{z})p_{j}(\beta)p_{k}(U)} \tag{21}\]  
where \(p_{i}(H_{1/3},T_{z})\), \(p_{j}(\beta)\) and \(p_{k}(U)\) are the probability of each wave state, wave heading angle and navigational speed, respectively;\(f_{0}^{ijk}(x)\) and \(F_{0}^{ijk}(x)\) are the probability density function and cumulative distribution function for each specified short-term condition, respectively; \(n_{0}^{ijk}\) is the cycle times of alternate loads within unit time period and it differs for each short-term condition, and it is obtained by:  
\[n_{0}^{ijk}=\frac{n_{ijk}}{T_{ijk}} \tag{22}\]  
where \(n_{ijk}\) is the cycle number of motion or load signal within the whole simulation time \(T_{ijk}\).  
The probability that the peak value exceed a specified value \(x_{a}\) can be obtained by:  
\[P\{X\geq x_{a}\}=\int_{x_{a}}^{\infty}f(x)\mathrm{d}x=1-F(x_{a})=\frac{1}{n} \tag{23}\]  
where the probability of exceedance \(P\) is the reciprocal of the cycle number \(n\), i.e., the number of waves encountered of a ship during the return period. The value of \(n\) is usually set at \(10^{8}\) for determination of the long-term design loads.