Page 176 - Pipelines and Risers

Force Model and Wave Fatigue I49

The time domain approach is to construct a time history of the irregular sea surface from a
wave spectrum &(a). Given, such a spectrum, the velocity and acceleration of water particles
given by the linear wave theory are:
U(Z) = iq J-ccls(qt +0, )
I=-.
;(I) = eq2J-sin(iT,t+0,)
I--"

where:
1
is
shh(m=-ss,(F) the wave height spectrum,
4
Bi is the phase angle uniformly distributed from 0 to 2%.
Given the above equations, a time series of velocity and acceleration can be constructed. The
span motion can then be analyzed in the time domain to obtain a time history of the response.

Before Equation (10.9) can be solved it is necessary to recast it, because the numerical
differential equation solver used only handles first order ordinary differential equations. By
introducing a new variable the equations become:
-2"
0)
dz
A- (10.10)
dz
d'Z,(t) &
-=. (10.11)
dzz dz

(10.12)

Equations (10.10) and (10.12) are solved to obtain Z,,(t). The pipe movement is then given by
Equation (10.8).

The spectrum of the pipe response is then calculated from the response time history by
Fourier Transformation. The advantage of the time history simulation is that non-linearity's in
the loading and response may correctly be taken into account. However, the calculation of the
transfer function also involves a linearisation process that is basically only appropriate for the
sea-state for which the simulation was done.