maximum absolute value and the maximum double amplitude (i。e。 peak-to-trough value)。

4。1。Quasi-static analyses

Initially, a series of quasi-static analyses was perfor- med, ignoring dynamic effects。 These highlight some  of

the important features of jack-up loading and serve as a basis for subsequent dynamic amplification comparisons。 Fig。 5 shows the variation of lateral hull displacements with wave period for the 13。0 m wave, in terms of both maximum and double amplitude values。 As would be expected, in each case the largest displacements occur for the rig with pinned feet and the smallest for the rig

Fig。 5。 Quasi-static hull displacements for a wave height of 13 m。 (a) Maximum values; (b) double  amplitudes。

with fixed feet, Model B footings giving intermediate results at all wave periods。 The maximum displacements, Fig。 5(a), show a minimum at a period of  approximately

11。0 s, while the double amplitudes, Fig。 5(b), have a roughly horizontal plateau between 7。6 and 9 s。 These features are directly related to the nature of the loading on the rig。 Fig。 6 shows the variation in total base shear and overturning moment with wave period。 It  can be seen that the variations of the hull displacements closely follow those of the overturning moments [Fig。   6(b)]。

The detailed form of these curves can be attributed to several interacting characteristics of the loading。 First, the antiphase period for this wave height is 7。6 s。 A rela- tively small response would therefore be expected at this period。 Second, as the wave period increases, both the particle velocity at the mean water level and the rate of decay with depth reduce。 At low periods this leads to a reduction in the total forces acting on the rig, but at higher periods the reduction in rate of depth decay leads to an increase in the total forces。 The change in vertical distribution of the loading with period also means that the overturning moments, Fig。 6(b), show a slightly dif- ferent trend to the horizontal forces, Fig。   6(a)。

Displacement results for the 6。0 m wave (not    shown

Fig。 6。 Variation of total rig loads for a wave height of 13 m。 (a) Total base shear; (b) total overturning   moment。

here) are dominated by the reinforcement effect of waves arriving in phase at the windward and leeward legs。 Both the maximum displacements and the double amplitudes exhibit clear peaks at the reinforcement period of 5。4 s。 As with the higher period range, the pinned footing case always yields the largest displacements and the fixed footing case the  smallest。

In general, the quasi-static LGMs (Fig。 7), show a reasonable degree of correlation with the hull displace- ments。 For the pinned footing case, there is very close agreement。 For the Model B and fixed cases, the rate of increase of LGMs at high periods is rather lower than for the hull displacements。 This effect is thought to be related to the way that the moments are distributed between the legs; the relationship between the windward and leeward LGMs is slightly non-linear under quasi- static loading。

4。2。Dynamic analyses

4。2。1。Hull displacements and lower guide  moments

Fig。 8 shows the maximum hull displacements under dynamic loading for 13 m waves。 It shows that,    within

Fig。 7。 Quasi-static maximum LGMs (windward leg) for a  wave height of 13 m。

Fig。 8。    Dynamic maximum hull displacements for 13 m  waves。

the period range considered, significant dynamic ampli- fication occurs due to wave loading both at the rig’s natural period and at twice its natural period。 For example, for the pinned case resonant peaks can be seen at wave periods of 8。5 and 17 s。 The second peak occurs because a 17 s wave has significant energy in the second harmonic, at 8。5 s, arising from two sources。 First, the Stokes’ fifth order wave theory includes a component at the second harmonic; in higher order wave theories this component may be reduced, though it would still be sig- nificant。 Second, the current velocity, when combined with the wave particle velocity, introduces a second har- monic into the drag term of the Morison equation; analy- ses performed with zero current velocity suggest this lat- ter effect is comparatively small [1]。 Similar results (not shown here) were obtained for 6 m   waves。