Appendix A: Hydrodynamic load model 1. Morison’s generalized equation The Morison generalized equation is an empirical formulation used for calculation of hydrodynamic loads on slender structures, i.e. slender with regard to wavelength and wave height. RIFLEX applies the following equations for the total hydrodynamic force per unit length normal to and tangential to the structure. \[dF_N=(f_N^{FK}\rho A\dot {u}_N+m_a^N\dot {u}_N-m_a^N\ddot {x}_N+\frac{1}{2}\rho C_d^N(u_N-\dot {x}_N)|u_N-\dot {x}_N|dl)\] \[dF_T=(f_T^{FK}\rho A\dot {u}_T+m_a^T\dot {u}_T-m_a^T\ddot {x}_T+\frac{1}{2}\rho C_d^T(u_T-\dot {x}_T)|u_T-\dot {x}_T|dl)\] where \(\mathrm {\rho }\) : water density \(\mathrm {A}\) : external area of the structural column \(\mathrm {m_A^N}\) ,\(\mathrm {m_A^T}\): added mass in normal and tangential directions \(\mathrm {C_{D}^{N}}\),\(\mathrm {C_D^T}\) : drag coefficient in normal and tangential directions \(\mathrm {u_N}\),\(\mathrm {u_T}\) : wave particle velocity in normal and tangential directions \(\mathrm {\dot {u}_N}\),\(\mathrm {\dot {u}_T}\) : wave particle acceleration in normal and tangential directions \(\mathrm {\dot {x}_N}\),\(\mathrm {\dot {x}_T}\) : structural velocity in normal and tangential directions \(\mathrm {\ddot {x}_N}\),\(\mathrm {\ddot {x}_T}\) : structural acceleration in normal and tangential directions The first term in the in Equation (1) and Equation (2) may be associated with Froude-Krylov forces. i.e. forces based on undisturbed pressure acting on the volume as if the structure was not present. The second term may be associated with forces from the diffracted pressure due to the presence of the structure restrained from oscillating. The third term may be associated with radiation forces from pressure that occurs when the structure is forced to oscillate. The fourth term expresses the quadratic drag force. \(\mathrm {f_N^{FK}}\) is a scaling factor for Froude-Krylov force acting in normal direction to the structure. It is introduced to extend the region of application of Morison’s equation, i.e. allow for larger structural diameter versus wavelength. \(\mathrm {f_T^{FK}}\) is a scaling factor used to activate/deactivate the Froude-Krylov force acting in the tangential direction of the structure. The Morison’s equation assumes the structure to be surrounded by water. If the structure penetrates the water surface a correction to the tangential loads must be introduced. This is equivalent to integrating the buoyancy load to the instantaneous wave elevation and the wave inertia loads (term 1 and 2 in Equation (1) and Equation (2)) to the still water level. The forces based on Morison’s equation are applied as distributed element loads. The loads at the nodes are calculated using lumped or consistent load formulation, as chosen by the user. 2. MacCamy-Fuchs load model The MacCamy-Fuchs method may be used to provide acceptable hydrodynamic forces outside the region for which Morison’s equation is valid. Note that the requirement for slender structural elements for applying beam theory still applies. For large diameter circular columns, diffraction effects become important in short waves. The MacCamy-Fuchs analytical solution for first order diffraction for a vertical surface-piercing cylinder has been implemented as pre-generated wave forces calculated before the dynamic simulation using the static coordinates. This approach is applicable for finite water depth and gives depth- and frequency-dependent wave loads up to the still water level. For linear wave theory, the results are exact for a pile whose diameter is much greater than the wave height. The solution is given for an earth fixed pile with constant diameter but is assumed to be applicable for cases where the diameter does not change too rapidly. The local diameter, vertical position, and components of the wave potential are the only required input to the force computation. For elements with MacCamy-Fuchs type loading, the formulation for the inertia and diffraction forces follows (MacCamy and Fuchs, 1954 and Dean and Dalrymple, 1991), with modifications for the irregular wave history. The force per unit length (\(\mathrm {dF}\)) for the column located horizontally in origin (at \(\mathrm {x=0}\)) for a given elevation \(\mathrm {z}\) is given by: \[dF(z)=\sum_{n=1}^{4}\frac{4\rho ga_n}{k_n}\frac{\cosh(k_n(z+h))}{\cosh(k_nh)}G\cos(\omega t-\epsilon_n-\alpha _n)\] where \[\tan_n=\frac{J_1^{'}(k_na)}{Y_1^{'}(k_na)}\] and \[G=\frac{1}{\sqrt{(J_1^{'}(k_na))^2+(Y_1^{'}(k_na))^2}}\] \(\mathrm {J_1^{'}}\) and \(\mathrm {Y_1^{'}}\) are the Bessel functions. The resulting \(\mathrm {C_M}\) and phase angle for a section with diameter 10 m is shown in figure Figure 1. For higher frequencies the mass coefficient is low, and the forces will be low compared to Morison. For lower frequencies the mass coefficient is 2, the same as with Morison and the two models are expected to give the same result. Note that the loads act in the horizontal plane. Time series for the MacCamy-Fuchs wave excitation loads are generated during pre-stochastic analysis and applied during dynamic analysis. Interpolation is applied if the simulation time step differs from the pre-generated interval Figure 1. Effective mass coefficient (blue) and force phase (red) To extend the use of MacCamy-Fuchs loads on bottom-fixed cylindrical monopiles to be applicable for floating single column systems a simple load model representing the radiation forces is implemented. The radiation loads are based on an added mass coefficient and a damping coefficient and included as: \[dF_H=(F_H^{MCF}-m_A^H\ddot {x}H-c^H\dot {x}_H)dz\] \(\mathrm {dF_H}\) is the force per unit length that includes the MacCamy-Fuchs and radiation contributions and that acts in the (global) horizonal plane \(\mathrm {F_H^{MCF}}\) is the MacCamy-Fuchs wave excitation load \(\mathrm {m_A^H}\) is the user specified added mass coefficient (Simplified radiation) \(\mathrm {c^H}\) is the user specified damping coefficient (Simplified radiation) \(\mathrm {\ddot {x}_H}\) is the structural acceleration in the horizontal plane \(\mathrm {\dot {x}_H}\) is the structural velocity in the horizontal plane The forces based on MacCamy-Fuchs load model including radiation contributions are calculated at the center of an element and distributed to the element ends (nodes). In the tangential direction the wave excitation forces are calculated in a similar manner as the corresponding load term in the generalized Morison equation. (\(\mathrm {dF_T}\)) The drag terms are calculated as described for the generalized Morison equation and act in the local element system.