Global static analysis (a) Hydrostatic stability

In Fig. 3.15, the terms needed for hydrostatic stability control are shown, where:

T

R = riser tension

T = anchorage tension

G = weight of platform including water ballast I

w = waterline moment of inertia.

For a unit rotation (1 rad), the centre of buoyancy moves a distance eB from the centre line.

(3.23)

where

 = specific weight of water (1.027 t/m3₎

g = gravity acceleration (9.81 m/s2₎

 = displacement (kN).

Moment equilibrium for a unit rotation relative to the anchorage level of the raft (notation see also Fig. 3.15)

M =  · (z_B + e_B - z_o) - G · (z_G - z_o) +  T · a - T_R · (z_R-Z_o) (3.24)

In (3.24) T is the change in tension of the anchorage lines for a unit rotation, expressed for a TLP as

(3.25)

where the axial stiffness term EA/L per tension line is introduced.

For the case of a free floating structure, as during construction, towing or installation, the riser and the anchorage contribution can be eliminated from (3.24). Then  = G and the stabilizing moment is

and

(3.27)

The Z

0 requirement is the stability criterion for a free-floating structure.

For a TLP in operation the tethers are the dominant items for hydrostatic stability.

(b) Static rigid body motion

The raft is assumed now as non-deformable and that the stiffness of the anchorage system determines the motion for given loads. As shown in Fig. 3.16 it is convenient to establish the equilibrium equations in the Cartesian co-ordinate system with the origin located in the centre of the anchorage system of the platform. A 6 x 6 stiffness relation is then established:

(3.28)

For a TLP the stiffness in the horizontal modes can be defined from the geometric stiffness

(3.29)

of the tethers:

where T

i is the pretension and Li length of tether number “i”.

The vertical stiffness in heave and roll will depend on the axial stiffness of the tethers, whereas the contribution of the water surface area is secondary. We get:

(3.30)

where EA/L is the axial stiffness of a tether and A

w the water surface area of the platform.

The expression in (3.28) is linear and does not take into account that the tensile forces in the anchorages change as a consequence of the platform motion. For a TLP where the stiffness of the tethers is dominant, the raft will follow an approximately circular path in the vertical plane for sideways motion. For a given horizontal displacement U, the increase in depth will be:

(3.31)

This increase in depth can be in the order of a few metres. This gives a buoyant force that, once more, alters the force in the tether.

(c) Static deformation of raft

This section deals with the deformation mode of the raft under the influence of static loads in order to obtain the sectional forces in the structure. The deformation pattern shown in Fig. 3.17

is a typical example for loads from waves and wind. Fig. 3.16 Global reference system

To obtain the deformations and the sectional forces one needs the stiffness of the pontoons, the shafts and the deck. It is also important to get the correct stiffness model of the boundaries between the pontoon and the shafts and between the shafts and the deck.

A method using hand calculations to obtain the sectional forces is shown below. An alternative method is to use a beam model in an element program, particularly when different load situations are to be analysed.

With reference to Fig. 3.18, the virtual deformation figure is chosen with the intention of calculating the moments at the boundaries of the pontoon. It is assumed here that we still have a simply supported connection between the top of the shaft and the deck, this assumption needs to be re-evaluated for each load situation. Given the pontoon moment as M_PON, and  as the virtual angle at a section with M_PON, the internal virtual work, including two pontoons, is:

Wi=4 · MPON · ß (3.32)

For loads parallel to the pontoon (0° or 90°):

The static load resultant from wind and/or current is assumed to act with a magnitude P at a height Z

p relative to the platform co-ordinate system. It is also assumed that due to P, the

change of the anchorage force will be T for each shaft. For a TLP where the axial stiffness is more dominant, then:

(3.33)

The outer virtual work is given as:

(3.34)

See also Fig. 3.18 for notations.

The relation between  and  is given from the geometry of Fig. 3.18:

(3.35)

By comparing (5.10, 5.12 and 5.13), the bending moment at the boundary of the pontoon is:

(3.36)

The relation above is evolved for a static load parallel to the actual pontoon. For a different load direction it is advantageous to use a simple frame program.

The moment diagram over the pontoon for the case shown in Fig. 3.18 would be linear with tension on the opposite side of the two pontoons and nil bending moment at the centre of the pontoon. The shear force from wind/current is constant along the pontoon.

By using the virtual work as mentioned above, it is vital to include all the loads contributing to the global equilibrium model, refer to contribution of T.

(d) Residual sectional forces

Eccentric positioning of the deck load during the mating operation generally leads to the need for jacking operations to reduce the deformations and the sectional forces in the flexible raft; see Fig. 3.12. This and the simultaneous deballasting of the raft after mating leads to permanent stresses in the raft which must be accounted for in design.

The residual sectional forces can be calculated by hand or alternatively by using a beam model in a computer. In this case, it is necessary to model the shaft/deck connection with a finite element model more accurately than is possible with hand calculations. A local shell element model combined with a beam model of the lower part of the raft can be used.

(e) Uneven ballast

Fig. 3.19 illustrates the situation with uneven ballasting for a doubly symmetrical raft. Two diagonally opposite shafts have more ballast water whereas the two others have correspondingly less. The total ballast and the draft are therefore correct, but the internal redistribution introduces stresses in the raft.

By having the modal amplitude as the difference in the deflection between the two sets of shafts, the stiffness of the model would be:

(3.37)

where EI

PON = sectional stiffness about the horizontal axis of the pontoon

L

PON = effective length of the pontoon, distance between the boundaries.

For a catenary anchored structure, the raft will carry the entire uneven load. For a TLP on the other hand, a portion of the load will be carried by the tether axial stiffness. To find the relative distribution between the tethers and the raft, a comparison is made between the raft modal stiffness in (3.37) and the modal stiffness of the tethers.

(3.38)

where EA

TETH= sectional stiffness of tethers for each shaft

For a TLP, the raft will carry most of the uneven load since the modal stiffness in (3.37) is 10 to 20 times larger than the contribution of the tethers (3.38).

The bending moment at the end of the pontoon is

(3.39)

where

P = the additional ballast per shaft.

From (3.39) the bending stresses in the pontoons can be calculated.