# Influence of Linear Springing on the Fatigue Damage of Ultra Large Ore Carriers
## 2 Method of Solution for Hydroelasticity  
The seminal and the most representative hydroelasticity method to numerically simulate springing was done by [PERSON] and [PERSON] [14]. The hull model is represented by a Timoshenko beam and the hydrodynamics by the strip theory [15]. In order to investigate the behavior of non-beam like structures, three-dimensional (3-D) hydroelasticity methods were devised by [PERSON] [16] and also by [PERSON] and [PERSON] [17]. In the early 1990s, hydroelasticity theo MV [PERSON] [18; 19]. The 3-D hydroelasticity method has now been widely applied in the analyry was applied to investigate structural failures of bulk carriers, such as the Onomichi Maru and OBsis of wave load and structural response of large ships and floating structures [20]. [PERSON] et al. [21] studied the motion and the load response of a 550,000 DWT ore carrier, both numerically and experimentally. The hydroelasticity theory and experiment were generally in good agreement, except that the rigid-body theory overestimates responses in the low frequency range. The hydroelasticity theory is more preferable when investigating the elastic effect on the wave load of large ships. By taking a similar approach [22], a linear 3-D hydroelasticity method has been implemented in this study and applied to analyze the contribution of springing to fatigue damage to structures [10; 21]. The free-surface Green's function is adopted to satisfy the linear free-surface condition, and the speed effect is accounted for by means of encounter frequency. In order to calculate the motion and wave load response of the flexible structures, the diffraction potential \(\varphi_{d}\) and the radiation potential \(\varphi_{k}\) are solved with the generalized fluid-structure coupling boundary conditions [23]:  
\[\left\{ \begin{array}{l}\frac{\partial\varphi_{d}}{\partial n}=-\frac{ \partial\varphi_{0}}{\partial n}\\ \frac{\partial\varphi}{\partial n}=\left[i\omega_{0}\mathbf{u_{r}}+(\mathbf{ W}\cdot\
abla)\mathbf{u_{r}}-(\mathbf{u_{r}}\cdot\
abla)\mathbf{W}\right]\cdot \mathbf{n}\end{array} \right. \tag{1}\]  
where \(\varphi_{0}\) is the incident wave potential, \(\omega_{0}\) the wave natural frequency, \(\mathbf{n}\) the vector normal to the surface, \(\mathbf{u_{r}}\) the central point displacement on the surface under the \(r\)th mode, and \(\mathbf{W}\) the constant velocity around the hull  
\[\mathbf{W}=-U\mathbf{i} \tag{2}\]  
where \(\mathbf{i}\) is the unit vector in the direction of the \(X\) axis and \(U\) is the navigation speed. The equation of motion for a flexible body travelling with forward speed \(U\) in regular deep-water waves can be written as  
\[\left[-\omega_{e}{}^{2}(\mathbf{a}+\mathbf{A})+i\omega_{e}(\mathbf{b}+\mathbf{ B})+(\mathbf{c}+\mathbf{C})\right]\mathbf{P_{r}}=\mathbf{F} \tag{3}\]  
where \(\omega_{e}\) is the encountered frequency; \(\mathbf{a}\),\(\mathbf{b}\), and \(\mathbf{c}\), of dimensions \(n\times n\), are the generalized masses, structural damping, and stiffness matrices, respectively; \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\), of dimensions \(n\times n\), are the generalized added inertia, hydrodynamic damping, and fluid restoring matrices, respectively; \(\mathbf{F}\), of dimension \(n\times 1\), is the generalized wave excitation vector, consisted of the contributions of both the incident wave and the diffraction; \(\mathbf{P_{r}}\), of dimension \(n\times 1\), is the principal coordinate vector. \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) can be calculated by  
\[\left\{ \begin{array}{l}A_{rk}=\frac{\text{Re}(H_{rk})}{\omega_{e}^{2}}\\ B_{rk}=-\frac{\text{Im}(H_{rk})}{\omega_{e}}\\ C_{rk}=\rho_{S}\iint\limits_{S_{0}}\mathbf{n}\cdot\mathbf{u_{r}}w_{k}ds- \iint\limits_{V_{b}}\rho_{b}\mathbf{u_{r}}\cdot(\mathbf{g_{s}}\times\mathbf{ \theta_{k}})dV\end{array} \right.