Page 227 - Dynamic Loading and Design of Structures

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mooring lines may be modelled by a simple spring-damper, or by a finite element model of
the line. Particular attention to the modelling of the leg-deck connections (SNAME, 1994) in
jack-ups is required. Pile foundations may be modelled by beam models and taking the
interaction between pile and soil into account by a continuum model of the soil; or by
representing the pile-soil behaviour by a simple spring-damper. A simplified multi-degree of
freedom boundary element method of the pile and soil, referred to as disk-cone model, has
proven to be computationally efficient (Wolf, 1994 and Emami Azadi, 1998). Mat or gravity
foundations can normally be well represented by a spring-damper model.
While the structure is normally assumed to have linear elastic properties when load effects
for component ultimate and fatigue limit states are determined, it may be necessary to account
for non-linearity in soil behaviour. However, when dynamic behaviour up to system collapse
is to be determined, non-linear material and geometric effects both of the structure, foundation
and soil would normally be required.
Mass is contributed by structural and contained mass as well as the added hydrodynamical
mass. For a slender cylinder the latter mass is usually taken to be that of the displaced water.
The added mass for large volume structures (e.g. caissons of floating gravity structures,
floating bridges) has to be determined by potential theory for the relevant modes of behaviour.
Particular attention should be paid to structural components which are close to the surface,
relative to their size. Ogilvie (1963) and Vugts (1970) give data for an infinite cylinder
moving horizontally at a certain distance below the water surface. Yeung (1989) determined
added mass for a vertical cylinder, and an infinite cross-section shaped like a ship moving in
the water surface. The added mass is frequency dependent.
Damping may be contributed by the structure, water and soil (rock) and is subject to
significant uncertainties. Structural damping (Barltrop and Adams, 1991) in a welded steel
structure may be of the order of 0.2–0.5 per cent of critical, and for concrete structures which
are stressed so that microcracks occur, it may be of the order of 0.5–1.5 per cent (Langen et
al., 1997). Structural damping of platforms or submerged bridges may be about 1 per cent
with pure structural modes of vibration.
Hydrodynamic damping stems from generation of waves (radiation damping) as well as
from friction drag damping. The first source is determined from potential theory and is given
for the special cases mentioned above in Ogilvie (1963), Vugts (1970) and Yeung (1989); it
exhibits strong dependence on frequency and submergence. For significant drag damping to
occur, vortex shedding must take place. The drag damping will be small if the KC number is
below, say, 2. Hence, drag damping will be small for large diameter vertical columns in
platforms and submerged bridges. The corresponding damping ratio may be less than 0.1 per
cent. Similarly, potential (radiation) damping is found to be relatively small compared with
drag damping for platform structures consisting of slender members. For floating bridges
wave difference frequency (slow drift) excitation may be of importance. Both drag damping
and second order (slow drift) potential damping are quite small at the excitation frequencies.