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KAZIAN GLOBAL SCHOOL OF BUSINESS MANAGEMENT

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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, to situations 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 and operator. 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 of the ship’s weight (
) 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. The consequence 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
sin
, 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 a circular 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
sin
. The only practical hull forms satisfying these conditions are circular 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
sinMNBGBMGZM
r
where
sinBM
: form stability
sinBG
: weight stability
sinMN
: residual stability
N is known as the prometacentre.

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Figure 7.1. Transverse stability 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 displacements over 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
sinKGKZGZ
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 and deckhouses above the weather deck are normally assumed to be
nonwatertight. Any actual watertightness of such structures, maintained by the proper closure
W
G
M
B
B
1
Z
N
K
Z
R

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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, to situations 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 and operator. 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 of the ship’s weight () 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 degr ...
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