Page 215 - Dynamic Loading and Design of Structures

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height to structural diameter remains sufficiently small. According to potential theory, the
pressure distribution and the corresponding forces can be calculated from the velocity
potential, as discussed in Section 5.3.3.
When the wave acts upon a structure, the latter will be set in motion, which will set up
waves radiating away from it. Reaction forces are then set up in the fluid that are proportional
to acceleration and velocity of the structure, respectively. These are inertia (added mass) and
potential damping forces due to wave generation. In addition, viscous (drag) forces are set up.
This issue is treated in Section 5.3.4. Finally, Section 5.3.5 deals with particular transient
wave loading phenomena such as wave slamming and ringing.

5.3.2 Steady-state loading on slender structures

If the characteristic dimension (e.g. the diameter D of a circular structural component) is
(
small relative to the wavelength λi.e. D/λ< 0.2), there is little al-teration of the incident wave
when it passes the structure. The wave does not ‘see’ such a slender structure: as diffraction
and reflection phenomena are negligible, the structure is said to be ‘hydrodynamically
transparent’. With relatively small dimensions, local variations of particle velocity and
acceleration in the region of the structural element are small enough to be ignored, and values
are calculated at the position of the structural element as a whole.
For slender members which are fixed the force per unit length qn, normal to the member, is
most often calculated by the extended, empirical Morison formula (e.g. Clauss et al., 1991):

(5.28)

where ρis the density of the fluid, CM and CD are the inertia and drag force coefficients,
respectively, and v and a are, respectively, the wave particle velocity and acceleration
n
n
perpendicular to the member. dA. and are the exposed area and displaced water of unit
length. For a circular member, with a diameter D,
The first term results from the FK and hydrodynamic mass force while the second
terms in eqn (5.28) is due to the viscous drag term and downstream wake.
The vn and an for a design wave are obtained directly from the kinematics for a regular
wave. In the case of random waves, v and a are obtained by superimposing the kinematics
n
n
for all regular waves that constitute the random wave history.
For vartical cylander in deep water, the total integrated force q and q are equal for a wave
D
I
height to diameter ratio of about 10.
A crucial issue in applying Morison’s equation is the determination of CD and CM·
Extensive data from laboratory experiments indicate a general range of 0.6 to 1.2 for C and
D
1.2 to 2.0 for C , depending upon flow conditions (as measured
M