Model for global response analysis

This section deals with a simplified model for global response analysis of gravity base concrete structures, which combines the contribution of the rigid element displacement of the structure and the beam effect of the shafts. The actual load effects are displacement and acceleration of the deck, and alternatively, beam moment and shear at the base of the shaft.

Fig. 3.4 points to several factors that need to be evaluated prior to running the global analysis model. These factors include: stiffness characteristics, mass motion and soil damping, as well as the deformation characteristics of the caisson cell walls that influence the degree of clamping at the base of shaft. The deck connection to the top of shafts affects also the moment and the shear force in the shafts. A total understanding of load distribution in the deck and the structure is necessary. Interaction with the surrounding water shall be accounted for. Fig. 3.5

There are now two ways to perform a simplified global analysis: either by a simplified element model (beam elements for shafts and shell elements in the caisson), or by hand calculations.

The following procedure gives a rough indication of these approaches:

A. The analyses should include all critical phases such as construction, towing, installation and operation.

B. For each phase an eigenvalue analysis is performed with due consideration of water mass.

C. If the natural period of the structure is substantially below the load period interval, a quasi-static calculation is performed by neglecting the mass contribution.

D. When masses are believed to influence the behaviour of a slender structure they are taken into account.

Further illustrations are based on simple hand calculations. Fig. 3.4 Simplified model

Simplified equations for one degree of freedom: Modal stiffness (3.3) Modal mass: (3.4) Eigenfrequency: (3.5) where
(x) = assumed deformation
,xx = curvature

m_s = mass of structure including ballast water m_a = additional mass from surrounding water.

a. Submerged phase for deck connection

b. Operation phase

The two degrees of freedom in the calculations are the displacement  of the shafts and rigid body rotation  of the caisson.

Fig. 3.6 shows a model of the submerged phase just prior to coupling the deck to the structure and also a model in the operation phase. In both models the shafts are assumed to be clamped at the top of the caisson. For the submerged phase, it is necessary to compare the natural frequency with the wave period at the actual location. In protected water, such as in a fjord, the values are somewhere between 2 and 6 seconds. Clement weather is needed for coupling operations.

In the operation phase the composite action of the deck and the structure is considered. In Fig. 3.6b a simply supported connection is indicated between the deck and the shaft (situation just after coupling). A fixed connection is then established by stressing cables and grouting the space between the deck panels and the shaft. An indication of the deck stiffness compared to the shaft stiffness can be obtained by calculating the modal stiffness in the same mode for the deck and the shaft.

In Fig. 3.7 an alternative global model is shown, where the caisson is assumed to be stiff, but the stiffness, the mass and the damping are included for the ground.

The general equations for the elements in modal stiffness, mass and damping are:

(3.6)

(3.7)

(3.8)

where: m

s = mass of structure including ballast water

m

a = additional mass of surrounding water

C

k = damping in structure

c

a = damping from surrounding water.

In relation to equations in Fig. 3.6, Fig. 3.7 includes rigid body displacement of shafts and caisson. It is appropriate to use a two-degrees of freedom system, where the modal amplitudes are, for example, the rotation of the caisson and the horizontal displacement of the top of shafts. The displacement pattern with two degrees of freedom may be described by the global modes (see Fig. 3.8):

u(x) =  · y for the caisson

Fig. 3.8 Displacement pattern The stiffness and mass calculations are then

Stiffness:

Mass: (3.12)

(3.11) (3.10) (3.9)

Horizontal displacement:

u(x)= · _(x)+ · _(x) (3.16) where _(x) and _(x) are the modal functions.

From this comes a 2 x 2 system of stiffness, mass and damping.

Two eigenvalues are obtained from the condition

det (K–2_{M)=0 (3.17)}

and the equivalent eigenvector from

(K-_i2 _{M) X}

i=0 (i=1,2) (3.18)

Simplified calculations of eigenvalues may be done by hand. An alternative method is to use a beam program that is merged to a corresponding element model of the deck. Fig. 3.9 shows such an analysis of the Condeep platform Sleipner A.

Eigenvalues can be used to study the sensitivity of the structure with regard to assumptions made in the analysis. In Fig. 3.10 a sensitivity study has been performed of eigenvalues with regard to variation in deck stiffness, ground stiffness and structure stiffness.

Eigenvectors from the calculations above are the modal displacement pattern. A global load analysis with a chosen governing wave can be calculated by hand by “placing” the wave load distribution in the modes above. The equations can be solved for each mode. The load effect is then increased by multiplying it by a dynamic factor.

If the inertia forces are substantial, the acceleration is computed for each mode. If several eigenforms are used, the total acceleration is calculated as the contribution from each individual period where the phase angles are considered. In most cases, the eigenperiods are so short with respect to the wave period that all the modes can be assumed to be in phase with the loading. The water mass is considered in the inertia forces.

Fig. 3.10 Eigenperiod as a function of flexibility