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
M r  GZ  BM  BG  MNsin 
where
BM sin 
BG sin 
MN sin 
: form stability
: weight stability
: residual stability
N is known as the prometacentre.
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N


M
G
Z

B
W
R
B1
K
Z
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
GZ  KZ  KG sin 
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
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
of watertight doors, will provide a margin of safety of additional intact stability beyond that
indicated by the cross curves at angles of heel that immerse the structures in question.
Adjustments are made to account for the volumes and moments of immersed appendages suc
as rudder, propellers, sonar domes, etc., and freely flooding spaces like large seachests.
Figure 7.2. Typical cross curves of stability
7.3. Statical Stability Curve
Cross curves of stability are a convenient form in which to store the information necessary to
determine the large angle stability characteristics of a ship at any displacment, but at a single
assumed position of the centre of gravity. For the ship operator as well as for the naval architect
during the design process, what is needed is a determination of the righting arms or moments of
a ship heeled to any angle while in a given loading condition at one displacement and with its
centre of gravity at a position different from the one that was assumed in preparing the cross
curves. The statical stability curve in which righting arms are plotted against angle of heel is the
appropriate format.
By combining weight (both magnitude and CG) with the hydrostatic properties, righting arms are
produced and the stability of the ship can be determined. Typically, righting arms are plotted for
a range of heel angles where the heel is assumed to be induced by a moment about the
longitudinal axis. A particular moment, such as that caused by the wind at a given speed, may
also be imposed. Further analysis of the properties of this righting arm curve lead to a formal
assessment of the vessel's stability. The ability to derive righting arms for inclinations in other
directions is also useful for some types of ships.
The statical stability curve is determined by picking values of the righting arms from the cross
curves at the appropriate displacement and correcting the values thus obtained to reflect the
actuaş position of the centre of gravity. Two corrections may be required; one for the height of G
above the keel (KG) and one for the distance of G off centreline (TCG). The vertical correction
is almost always necessary, since it would be only by gratuitous coincidence that a ship’s KG in
a given loading condition would be exactly at the pole point chosen arbitrarily when the cross
curves were prepared. The lateral correction is, by contrast, rarely needed in routine loading
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conditions because good practice in loading a ship requires that the centre of gravity be on
centreline so that the ship will float upright.
It is clear that all righting arms will be smaller than those plotted in the cross curves,
because the actual centre of gravity is higher than the assumed centre of gravity, the keel, K.
Thus stability at large angles decreases as G rises, just as initial stability measured by GM does.
The conventional plotting of a statical stability curve is shown in Figure 7.3. Since the
centre of gravity is on centreline, the direction of assumed heel (staboard or port) is immaterial,
because the centreline symmetry of the hull would cause the centre of buoyancy to assume
symetrically corresponding positions on either side at any given angle of heel. Although the
curve can be plotted for all angles of heel for which cross curves have been determined, it is
typically terminated where the GZs become negative, that is, where righting arms change to
capsizing arms.
To have a full understanding of intact ship stability, we must know not only how a statical
stability curve is determined, but also why it is shaped as shown, and what significance is to be
attached to its typical features.
The initial portion of the statical stability curve (the first 7-10 degrees) must be consistent with
the measure of initial stability, that is, the metacentric height (GM).
GZ  GM sin 
As the heel angle approaches zero, sin    . Thus for the small angles of heel we may write
GZ  GM  
Therefore the metacentric height (GM) is a measure of the slope of a staical stability curve at the
origin and should always be used as an aid to plotting the curve, by running the curve in tangent
to the straight line at the origin. At an angle of 1 radian (equal to 180/  , or 57.3 degrees) the
straight line passes through the value GZ=GM. Thus, as is shown in Figure 7.3 , if GM is laid
out as an ordinate at 57.3 degrees and that point is connected to the origin by a straight line, the
statical stability curve will approach that line asymptotically as it approaches the origin.
GZ (m)
4
maximum GZ
3
GM
2
1
range of
initial
stability
1 radian = 57.3 deg
angle of maximum
stability
10
20
40
30
range of stability
50
60
70 Heel angle (deg)
Figure 7.3. Typical static stability curve
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For the great majority of ship hull forms, the statical stability curve from its initial path with
increasing slope so that it rises above the tangent line as the angle of heel increases. Eventually,
as the angle of heel increase, a point is reached at which deck edge immersion takes place. In
fact, since a ship’s section shapes vary from bow to stern, deck edge immersion is not a sudden
occurrence along the entire length of the ship, but take place gradually over a range of heel
angles. But the general trend is accompanied by a much reduced growth in GZ, and thus an
inflection point in the statical stability curve. This steadily decreasing slope of the statical
stability curve beyond deck edge immersion leads to the peak of the curve, and ultimately to its
rapid plunge beyond the peak.
This effect of deck edge immersion on large angle stability has important consequences for both
the design naval architect and the ship operator. The design implications are that a ship designed
to have a small freeboard may develop inadequate righting arms and moments at large angles
because deck edge immersion and the peak of the GZ curve will occur at relatively small angles
of heel.
To avoid this problem, low-freeboard ships must be designed with relatively large metacentric
heights, because the large initial slope of the statical stability curve will tend to ensure that
adequate righting moments will be achieved in spite of the small angle at which the curve
reaches its peak.
The peak of a statical stability curve identifies two quantities that are important in evaluating the
overall stability of a ship. They are the maximum righting arm and the angle of maximum
stability. The importance of the maximum righting arm (GZmax) is that the product of the
displacement and GZmax is the maximum steady heeling moment that the ship can experience
without capsizing.
Beyond the angle of maximum stability, righting arms decrease, often more rapidly than they
had increased up to that point. The rapid decrease ultimately leads to the point at which GZ
becomes zero, and the curve recrosses the axis. The angle at which this occurs is the angle of
vanishing stability, because thereafter the GZs are negative. That is, they are capsizing or
upsetting arms, rather than righting arms. Any ship that inclines beyond its angle of maximum
stability will capsize, regardless of the cause of the inclination or its duration. The typical
statical stability curve like that in Figure 7.3 crosses the horizontal axis at two angles of
inclination, each of which represents a condition of static equilibrium, since GZ equals zero. The
first crossing (zero heel angle) is a stable equilibrium condition, because temporary inclinations
to larger angles create righting moments that will restore the ship to the equilibrium angle when
the cause of inclination is removed. The second crossing is at the angle of vanishing stability and
represents a unstable equilibrium condition, because temporary inclinations to larger angles
create upsetting moments that will cause the ship incline away from that equilibrium condition
when the cause of inclination is removed. The range of heel angles between the two crossings is
called the range of stability.
As a practical matter, one must be careful not to depend too much on any ship’s ability to
recover from angles of inclination beyond its angle of maximum stability because the cross
curves of stability were determined on the assumption of perfect watertight integrity of the
weather deck. This assumption is incorrect for the great majority of ships. All the ships are
provided in deck for cargo hatches, access to spaces below decks, ventillation, piping, etc.
Properly designed deck penetrations are made weathertight, but few of them can be made truly
watertight, and there is always the possibilty of human error, that is , leaving doors and hatches
open that should be tightly closed again heavy weather. At angles of heel that immerse part of
the deck, the possibility always exists that water will be shipped through such openings. This
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event is called downflooding, and the smallest angle of heel at which it can occur is known as
the downflooding angle. If large quantities of water enter the ship, the statical stability curve is
no longer correct; the ship’s loading has changed. Therefore at any angle of heel greater than the
downflooding angle, a statical stability curve must be considered invalid. For large ships,
downflooding is rarely considered to be a critical issue, but for smaller vessels, fishing boats in
particular, the range of stability should be assumed to be terminated at the downflooding angle
rather than the angle of vanishing stability.
7.4. Stability of a Body of Revolution
Although the body of revolution is not a realistic form for surface ships, it is instructive to
examine the behaviour of such a vessel when it is heeled to large angles. Imagine a vessel, all of
whose sections are circular, loaded such that it floats with its axis horizontal, but with its centre
of gravity below its axis. It is immaterial whether all sections are of the same diameter (a circular
cylinder) or not ( a body of revolution for eaxmple, shaped like a cigar or a football). Figure 7.4
depicts this body floating at various angles of heel, the upright condition (   0 0) representing
the position of stable equilibrium, in which G is directly below the centre of the circle. It is
apparenr, as each of the positions shown are examined, that, at any angle of heel, the buoyant
force acts throug the centre of the circle, which is thus the transverse metacentre M. The righting
arms GZ are, for all angles of heel, equal to GM sin  . Note that the righting arms increase to a
maximum value equal to GM at 90 degrees, then decrease again to zero in the inverted position,
or 180 degree heel. This upside-down position is therefore one of equilibrium (GZ=0), but thr
slightest disturbance from it in either direction will cause the vessel to rotate until it returns to its
original upright condition. Thus it is a position of unstable equilibrium.
K
K
M
Z
G
B
K
M
G
B
K
G
M
B
G
Z
M
B
K
G
M
B
Z
M
G
B
K
Figure 7.4. Statical stability curve for a body of revolution
In this special case, the statical stability curve may also be described as a sine curve of amplitude
GM. Note that the same curve would result regardless of the draught (or displacement) of the
cylinder because KM is equal to the radius of the cylinder whatever amount is underwater.
Righting moments, of course, increase with displacement (or weight), even though righting arms
do not.
This statical stability curve shows that a body of revolution has, in one complete revolution, two
equilibrium angles (zero and 180 degrees), only one of which is stable (zero degrees). Thus such
a body will return to the upright condition from any angle of heel imposed on it externally.
The characteristic most strongly influencing the righting arms is the position of the centre of
gravity, that is, KG. If the centre of gravity is lowered, righting arms at all angles will get larger,
and, conversely, if G moves up, all righting arms will decrease. This relationship applies to ship
forms as well.
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7.5. Effect of KG on Righting Arm
Existing statical stability curves could be used for different loading conditions corresponding to
different values of the height of centre of gravity. For example consider two different loading
conditions corresponding to lower and higher value of KG, as shown in the following figure.
M


G1
G

Z
G2
B
Z1
Z2
W
R
B1
Z
K
Figure 7.5. Effect of vertical movement of KG on righting arm
When the cntre of gravity moves upwards to G1 the change in the righting arm is
G1 Z1  GZ  GG1 sin 
The modified righting arm curve will bes shown in the following figure
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GZ
GZ
GG1sin
G1Z1
GG1sin
Heel angle
Figure 7.6. Effect of vertical change of KG on righting arm
When the centre of gravity moves to G2, the new value of GZ is
G 2 Z 2  GZ  GG 2 sin 
Therefore, the change in GZ corresponding to a change in KG is
GZ new  GZ org  GG 1 sin 
The sign will depend on the position of the centre of gravity.
A horizantal shift of KG will always result in negative GM and heel.
GZ new  GZ org  GG 1 cos 
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M


G
G1

B
W
Z
Z1
R
B1
Z
K
Figure 7.7. Effect of horizontal change of KG on righting arm
GZ
GG1cos
GZ
GG1cos
G1Z1
Heel angle
Figure 7.8. Effect of horizontal change of KG on righting arm
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