Curvas cruzadas

CHAPTER 7
TRANSVERSE STABILITY AT LARGE ANGLES OF HEEL



The essence of stability calculations is finding the force couple between buoyancy and weight. This is the moment of force which a stable ship develops to counteract the overturning moments arising from external forces. Reliance on the metacentric height as a measure of transverse stability is limited, as described in Chapter 6, tosituations in which the ship heels to small angles from the upright, typically less than about 10 degrees. If the upsetting forces that act upon ships in service, such as those caused by wind, waves, cargo handling, and turning, could not produce inclinations larger than a few degrees, the study of metacentric, or initial transverse statical stability would be sufficient for both ship designer andoperator. However, ships can and do heel and roll to larger angles under the influence of large heeling moments. To ensure proper design and safe operation we must know how a ship behaves when heeled to large angles.

7.1. Righting Arm and Righting Moment

Whatever the angle of heel, the proper measure of a ship’s ability to return to upright is the righting moment, equal to the product ofthe ship’s weight ([pic]) and the righting arm (GZ), as shown in Figure 7.1. The difference between the small and large angles of heel is due to the fact that at large angles the buoyant force vector does not pass through the metacentre (M). The reason is that, as the angle of heel increases beyond a few degrees, the path of the centre of buoyancy (B) departs from a circular arc of radius BM. Theconsequence of this departure is that the righting arm is no longer related in any simple way to the metacentric height, that is, GZ is not equal to GM[pic], as it is in the case of very small angles of heel. In fact, no exact formula is known that relates GM to the righting arms GZ for large angles, except for the very restrictive class of hull forms for which the centre of buoyancy traces acircular path when the vessel heels to any angle. This will be the case only for spheres, circular cylinders, or bodies of revolution floating with their axis of symmetry parallel to the water surface. For such forms, the transverse metacentre lies on the axis of symmetry and the righting arms for all angles of heel are equal to GM[pic]. The only practical hull forms satisfying these conditions arecircular section pontoons and submarines whose hull forms are essentially bodies of revolution.

Once the righting arm, GZ, is determined for a given heel angle and loading condition the righting moment can be estimated as

[pic]

where

[pic] : form stability
[pic] : weight stability
[pic] : residual stability

N is known as the prometacentre.






Figure 7.1. Transversestability at large angles of heel



7.2. Cross Curves of Stability


The results of the righting arm calculations for a ship are plotted as a set of cross curves known as cross curves of stability. These curves are used to determine the length of the righting arm at any angle of inclination for a given displacement. A typical set of cross curves is shown in Figure 7.2. The range of displacementsover which cross curves have been determined is from the light ship displacement at the lower end to a displacement usually well above the load displacement, so that stability can be assessed at deep draughts associated with potential flooding situations.

Since the centre of gravity is a function of loading condition the basis of the cross curves is taken as a fixed point, such as the keel(K). Then the righting arm is

[pic]

In the preparation of cross curves of stability, certain assumptions have been made, as follows;

• The ship’s centre of gravity remains fixed at the pole point, or assumed centre of gravity, regardless of the angle of heel.
• The ship’s hull, consisting of the bottom, sides, and weather deck, is assumed to be perfectly watertight.
• Superstructures...