MacCamy-Fuchs Load Model

1. Description

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 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

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:

\[dF_H= (F_H^{MCF} - m_A^H\ddot{x}H-c^H \dot{x}_H )dz\]

  • \(dF_H\) is the force per unit length that includes the MacCamy-Fuchs and radiation contributions and that acts in the (global) horizonal plane

  • \(F_H^{MCF}\) is the MacCamy-Fuchs wave excitation load

  • \(m_A^H\) is the user specified added mass coefficient (Simplified radiation)

  • \(c^H\) is the user specified damping coefficient (Simplified radiation)

  • \(\ddot{x}_H\) is the structural acceleration in the horizontal plane

  • \(\dot{x}_H\) 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).

The tangential direction the wave excitation forces are calculated in a similar manner as the corresponding load term in the generalized Morison equation. (\(dF_T\)). The drag terms are calculated as described for the generalized Morison equation and act in the local element system.
Application of hydrodynamic wave loads for the different load models. Note that some loads are calculated and applied in the local element system while other are calculated and applied in the global vertical and horizontal directions.

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2. Theory

For details, see Theory - Load models - MacCamy-Fuchs with Quadratic Drag