from geometrical and P-A effects, since plasticity is not taken into account。 Jack-ups would not be expected to exhibit plastic behaviour within their working   range。

In the program the lattice legs are modelled by equiv- alent elastic beam elements。 The tangent stiffness matrix of these elements is derived using an Eulerian beam- column formulation based on that of Oran [3] which accounts fully for large deformations and P-A effects, with a modification to account for shear deformations [4]。 This approach gives accurate results using relatively few beam-column elements。 A consistent mass matrix (based on the initial linear shape functions) is used and structural damping is modelled using the Rayleigh approach, in which the damping matrix is taken as a linear combination of the mass and stiffness   matrices。

2。2。Foundation model

In most jack-up dynamic analyses, the spudcan foun- dations are modelled by pinned supports [5], i。e。 having infinite stiffness in the vertical and horizontal directions, but no rotational stiffness。 However, Bell [6] has shown that footing stiffness is highly dependent on embedment depth, and that spudcans introduce cross-coupling between the horizontal and rotational degrees of free- dom。 Martin [4] has investigated experimentally the yielding behaviour of spudcan footings on clay。 He developed a foundation model, known as Model B, which incorporates the dependence on embedment depth, and the necessary the cross-coupling between all the applied force components。 It is formulated as a work- hardening plasticity model and is capable of describing yielding and post-yield behaviour。 Detailed description of the model is beyond the scope of this paper, but it represents as realistically as is currently possible the behaviour of a spudcan on clay。 The program includes options to model the spudcan footings by pinned or fixed supports, or using Model  B。

2。3。Hydrodynamics

For hydrodynamic purposes the legs of the jack-up are idealised as equivalent cylinders, and the forces on these determined from the wave kinematics (see Section 2。5) and the extended Morison equation。 This accounts for the effects of hydrodynamic damping and added mass。 Relative motions of the leg and water are properly accounted for。 The equivalent dimensions, together with the drag and inertial coefficients used, are given  in Table 1。

2。4。Wave loading

All the analyses presented in this paper use Stokes’ fifth order wave theory to describe the water particle kin- ematics。 This theory approximates the wave   kinematics

Table 1

Main analysis parameters for representative  jack-up

Structure Single leg EI Single leg AE Single leg AsG 3 × 106   MNm2

120  × 103 MN

3。2 × 103  MN

Single leg mass 1。93 × 106  kg

Deck mass 16。1 × 106  kg

Structural damping 2% of critical in main sway   mode

Fundamental period (frequency) — linear  system

Fundamental period (frequency) — non-linear  system Pinned: Tn = 7 s (fn = 0。14 Hz) Model B: Tn = 5。6 s (fn = 0。18 Hz) Fixed: Tn = 3。4 s (fn = 0。29 Hz) Pinned: Tn = 8。3 s (fn = 0。12 Hz) Model B: Tn = 6。6 s (fn = 0。15 Hz) Fixed: Tn   = 4。6 s (fn   = 0。22 Hz)

Hydro-dynamics

Equivalent leg diameter De Equivalent leg area Ae Equivalent drag coefficient  Cde

2。14 m

3。5 m2 2。23

Equivalent inertia coefficient  Cme