The fundamental periods for fully linear behaviour were found by eigenvalue analysis, using stiffness matr- ices for the legs in their undeformed positions and with no axial loads。 For non-linear behaviour, estimates  of the fundamental periods were obtained by studying the responses to lateral impulse loads; the results are, of course, only indicative figures, since the period of oscil- lations will vary with the degree of non-linearity induced。 The periods in Table 1 show   that:

1。the inclusion of axial loads in the legs results in a significant reduction in the  sway  stiffness  of  the rig, and

2。in the absence of yielding of the foundations, the rig with Model B footings is significantly stiffer than that with pinned feet。

The soil properties used are typical of a North Sea

deep water environment。 The initial plastic penetration of the footings was found by performing a quasi-static analysis of the preloading phase for the Model B foot- ings。 Although only the Model B analyses require esti- mates of penetration depth, the same depth was used for the pinned and fixed footing analyses in order to  give the same total leg  length。

The loading direction (single leg to windward) has been chosen for convenience, and does not necessarily represent a worst case。 The mean water depth is towards the upper end of the depth range for existing rigs。 Most of the analyses presented here were performed using a fixed wave height H, to study the effect of variations in wave period T。 In addition, a series of analyses was performed with fixed period and the varying wave height。

4。Analyses at a fixed wave  height

The periods of most ocean waves are in the range 2– 20 s [11]。 However at the lower end of this range only very small wave heights are stable, and so these are of comparatively little interest to the rig designer。 In order to provide a wave height that can be applied across a wide period range, a value of 13。0 m was chosen for the fixed height computer simulations。 The corresponding period range was 5。8–20 s, the lower limit being the per- iod below which convergence difficulties are encoun- tered in the Stokes’ fifth order wave   formulation。

To give results at the lower periods associated with the fixed-footing model (see Table 1), a second series of fixed height analyses was performed, using a wave height of 6 m, the associated period range being 4。6–9 s。 Fig。 3 shows the variation of wavelength with period for these waves。 It can be seen that, for each wave height, there is a period at which the waves arrive at the windward and leeward legs in antiphase。 This results  in a reduced response, but is not strictly a cancellation per- iod, due to the presence of several harmonics in the wave loading function, and to the fact that there is a single leg to the  windward  side but  two  to leeward。  For  the 6 m wave there is a reinforcement period, at which the waves arrive at the windward and leeward legs in phase。 There are many structural response parameters that could be examined, but two of the most important are the lateral hull displacement and the moment at the leg/hull connection (the lower guide moment, or LGM)。 For the sake of brevity, most of the results presented here are in terms of hull displacement, since the LGMs generally follow a very similar trend; any slight discrepancies   are

highlighted below。

As can be seen in Fig。 4, the presence of steady wind and current loads causes rig oscillations to occur   about a non-zero mean value。 The salient features of rig response  are  therefore  presented  in  terms  of  both the

Fig。 3。    Variation of wavelength with period for waves of fixed   height。

Fig。 4。   Dynamic response of rig with pinned feet to a wave of height 13 m, period 17 s。