Page 220 - Dynamic Loading and Design of Structures

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floating structures and structural dynamics of bottom supported structures. In the latter case
the relative velocity term in eqn (5.31) should be used with caution. The amplitude of the
structures motion needs to be equal to the member diameter to set up the fluid flow for which
eqn (5.31) is valid. Otherwise, using eqn (5.31) may overestimate the damping and hence lead
to non-conservative load effects.
Analogous considerations apply to large volume structures (large cross-section dimension
relative to the wavelength). However, in that case the FK and added mass loads both need to
be determined by analytical or numerical methods, as mentioned in Section 5.3.3. The added
mass and damping contributions are then determined by introducing a potential Φ for each of
j
the six rigid body modes as well as possible flexible modes.
For the structures considered herein wave kinematics is commonly determined with
reference to the initial position of the structure. When motion amplitudes become large (i.e. of
the order of the wave amplitude), the position of the structure may be updated, when
excitation forces are determined.

5.3.5 Non-linear wave loading

Slender bodies
The drag force in Morison’s equation, eqn (5.31), is non-linear in particle velocity. The
particle velocity is proportional to wave height according to linear theory. Moreover, the fact
that the drag force is non-linear will introduce higher order harmonics in the force associated
with a regular, periodic wave. The drag and inertia force on, for example, a horizontal
member caused by a regular wave with particle velocity v =sin(ω) and acceleration
t
x
ax=cos(ωt) (Mo, 1983; Mo and Moan, 1984) are:

(5.32)

(5.33)

When a harmonic wave of finite height passes a structure, forces on a horizontal or a segment
of a vertical member in the ‘splash zone’ may vary in time as indicated in Figure 5.8. Clearly,
by expanding these forces in Fourier series, it is observed that there will be higher order
harmonic components in the overall excitation of the structure. This effect will be more
pronounced when drag forces are predominant because they attain their maxima at maximum
and minimum wave elevation. Also, drag forces are more important in an extreme seaway
than in a moderate one.
To illustrate this point more explicitly, consider a cylinder piercing the wave surface. When
the velocity is assumed to be constant above the MWL and equal to the velocity at the MWL
(vertical extrapolation in Section 5.2.2), the drag and