Hydrostatics

Introduction

For every floating body therefore for all offshore structures the buoyancy and the stability are critical to the feasibility of the design. The first requirement that the platform has sufficient buoyancy to carry the total weight of the structure itself and, in our case, the weight of the topside facilities (2x6 MW wind turbines, accommodation, substation, crane, research centre).

Methodology

Buoyancy is the upward force that the water applies to a body and, depending on the weight of the structure and the force, the body may float, sink or remain neutrally buoyant in the water [1].

The laws of the buoyancy can be summarised as follows:

The static forces acting on a floating platform in the water are gravity (the total weight of the structure) and buoyancy (the weight of the water displaced by the platform hull).

Figure 1. Stability reference points

Figure 2. Metacenter point [2]

An important quantity in the buoyancy analysis is the metacentric height (GM). The metacentric height is a measurement of the initial static stability of the floating body.

The equation described the metacentric height is:

GM=KB+BM-KG [4]

The above formula represents the available of the floating body GM, where:

The first thing needs to be calculated is the KB. KB can be calculated from the below equation:

KB=0.5DWL

Where DWL is equal to the draft of the structure (the distance from the Keel to the water plane).

Secondly BM should be calculated. BM is very important in buoyancy analysis and is equal to the moment of inertia around the water plane over the volume of displacement.

BM=I/V [4]

The moment of inertia can be calculated assuming rectangle beam from the equation bellow:

I=1/12 LB³

Where:

There are three particular stability conditions regarding the numerical value of metacentric height.

In order a floating body to be stable the numerical value of the metacentric height should be positive. This is the first floating condition. In case the metacentric height is larger that means that the floating body has better initial stability against overturning [2].

Afterwards and regarding the second floating condition the displaced weight of water should be equal to the total weight of the structure.

Analysis and Results

The buoyancy analysis accomplished for the needs of our project focused on the calculation of the metacentric height. The platform was modelled in AutoCad, then imported to MaxSurf to carry out the analysis, at which point some design simplifications were made. It was assumed that the platform was a full concrete box and the curves and slopes of the external cylinders were not included, as this was only a first pass analysis to determine the overall feasibility of the design.

Figure 3. Design layout in MaxSurf

As a result, volume of the platform’s concrete was marginally overestimated; however, this assumption will affect the analysis results by less than 2%.

Moreover, only the total weight of the structure was included in the analysis – the topside facilities total weight was estimated to be 4,000 to 5,000 tons, which is less than 1% of the total weight, therefore the model was simplified to be solely the concrete structure. As the design of the PSP allows the draft to be altered by varying the air volume in the cylinders, the draft was set to 10 m. Due to the solid deck, the centre of gravity was shifted slightly from the midline and assumed to be 20 m.

The variables imported in the software to run the analysis were the layout of the platform, the height, the density of concrete, the draft and the Centre of Gravity. The MaxSurf results are presented in the following table:

Conclusions

It is clearly seen that the metacentric height, GMt corrected, is positive:

GM=KB+BM-KG=5.0+50.7-20.0=35.7 m

As a result, the platform is in stable condition and the first requirement is fulfilled. Moreover, it is obvious that we are far from the neutral equilibrium. When a small angular displacement is applied the platform tends to return to its preliminary position.

The large metacentric height in this case justifies greater stability. The platform rolls with small roll angles and periods. The structure is considered more “stiff” in roll.

Less stable floating bodies have smaller metacentric heights and roll slowly with longer roll periods [5].

Afterwards assuming a full reinforced concrete box, the displaced weight of water is equal to 332,512 tons and fulfils the second requirement of the floating conditions.

Despite the fact that many assumptions made, the results are reasonable. However, for more accurate results in depth analysis is required.