Page 223 - Dynamic Loading and Design of Structures

Page 196

The purpose of the higher order theory is to approximate more accurately the boundary
conditions (i.e. the zero normal flow condition at the instantaneous position of the body and
the pressure condition at the free boundary). Such higher order excitation forces are
commonly derived by a perturbation method, with the following assumptions: variables, x
such as wave height, velocity potential, dynamic pressures and motions of the structure are
expanded in a series of a small perturbation parameters a.

(5.38)

(0)
where x represents the stillwater condition and x (1) corresponds to the first order (linear)
approximation. It is noted in particular that the different terms of quadratic velocity potential

are quadratic functions of the first order potentials

. Each of the quadratic potentials must satisfy the Laplace differential
equation and the boundary conditions at the free surface sea bottom and far field mentioned in
Section 5.2.2.
First order wave forces are then expressed by first order velocity potentials and first order
motions, taken care of by the equation of motion. Second order wave forces can then be
explicitly determined on the basis of the second order velocity potentials and first order
potentials as well as hydrodynamic pressures and motions.
Eatock Taylor and Hung (1987) calculated numerically the complete second order forces
on a cylinder.
Non-linear (second and higher order) wave forces generally are an order of magnitude less
than the first order (linear) forces. However, if the period of the wave force coincides with a
natural period, the effect of such forces can be large.
High-frequency horizontal forces on towers made up of slender members and vertical
forces on tension leg platform hulls may be of importance. Low-frequency horizontal (and
vertical) forces may be of importance to the motions of floating structures and tension-leg
platforms.

Ringing loads
Steep, high waves encountering structural components extending above the still water level
may cause non-linear transient loads and load effects. Figure 5.9 shows a measured irregular
wave profile and the corresponding horizontal forces for a short time sample involving a steep
wave. It is observed that a transient high frequency load occurs. Its amplitude is
approximately 20 per cent of the steady state amplitude. Structural responses to these actions
may be dynamically amplified and cause increased extreme response (ringing). Such transient
nonlinear actions may be important for structures consisting of large diameter shafts and
having natural periods in the range of 2 to 8 sec and started to receive serious attention in
connection with monotower, gravity base and tension leg platforms at the beginning of the
1990s. Ringing loads depend on the wave shape and particle kinematics close to the wave
surface and are highly non-linear, and it is generally difficult to distinguish impact/slamming
phenomena from higher order