1. By Mazlan Muslim, MEng,
UniKL MIMET

2.  In general, a rigid body is considered to be in a state
of static equilibrium when the resultants of all forces
and moments acting on the body are zero. In dealing
with static floating body stability, we are interested in
that state of equilibrium associated with the floating
body upright and at rest in a still liquid. In this ease,
the resultant of all gravity forces (weights) acting
downward and the resultant of the buoyancy forces
acting upward on the body are of equal magnitude
and are applied in the same vertical line.
 Stable Equilibrium
 If a floating body, initially at equilibrium, is disturbed
by an external moment, there will be a change in its
angular attitude. If upon removal of the external
moment, the body tends to return to its original
position, it is said to have been in stable equilibrium
and to have positive stability.

3.  Neutral Equilibrium
 If, on the other hand, a floating body that assumes a
displaced inclination because of an external moment
remains in that displaced position when the external
moment is removed, the body is said to have been in
neutral equilibrium and has neutral stability. A floating
cylindrical homogeneous log would be in neutral
equilibrium in heel.
 Unstable Equilibrium
 If, for a floating body displaced from its original angular
attitude, the displacement continues to increase in the
same direction after the moment is removed, it is said to
have been in unstable equilibrium and was initially
unstable. Note that there may be a situation in which the
body is stable with respect to "small" displacements and
unstable with respect to larger displacements from the
equilibrium position. This is a very common situation for a
ship, and we will consider cases of stability at small angles
of heel (initial stability) and at large angles separately.

4.  This chapter deals with the forces and moments acting on a ship
afloat in calm water. The forces consist primarily of gravity forces
(weights) and buoyancy forces. Therefore, equations are usually
developed using displacement, Δ, weight, W, and component
weights, w. In the "English" system, displacement, weights, and
buoyant forces are thus expressed in the familiar units of long
tons (or lb.). When using the International System of Units (SI), the
displacement or buoyancy force is still expressed as Δ=ρg∇, but
this is units of newtons which, for most ships, will be an
inconveniently large number. In order to deal with numbers of
more reasonable size, we may express displacement in
kilonewtons or meganewtons.
 A non-SI force unit, the "metric ton force," or "tonnef," is defined
as the force exerted by gravity on a mass of 1000 KG. If the
weight or displacement is expressed in tonnef, its numerical value
is approximately the same as the value in long tons, the unit
traditionally used for expressing weights and displacement in ship
work. Since the shipping and shipbuilding industries have a long
history of using long tons and are familiar with the numerical
values of weights and forces in these units, the tonnef (often
written as just tonne) has been and is still commonly used for
expressing weight and buoyancy. With this convention, righting
and heeling moments are then expressed in units of metric ton-meters,
t-m.

5.  The total weight, or displacement, of a ship can be
determined from the draft marks and curves of
form, as discussed in Geometry of Ships. The
position of the center of gravity (CG) may be
either calculated or determined experimentally.
Both methods are used when dealing with ships.
The weight and CG of a ship that has not yet been
launched can be established only by a weight
estimate, which is a summation of the estimated
weights and moments of all the various items that
make up the ship. In principle, all of the
component parts that make up the ship could be
weighed and recorded during the construction
process to arrive at a finished weight and CG, but
this is seldom done except for a few special craft
in which the weight and CG are extremely critical.
 After the ship is afloat, the weight and CG can be
accurately established by an inclining experiment.

6.  To calculate the position of the CG of any object, it is
assumed to be divided into a number of individual
components or particles, the weight and CG of each being
known. The moment of each particle is calculated by
multiplying its weight by its distance from a reference
plane, the weights and moments of all the particles added,
and the total moment divided by the total weight of all
particles, W The result is the distance of the CG from the
reference plane.
 The location of the CG is completely determined when its
distance from each of three planes has been established. In
ship calculations, the three reference planes generally used
are a horizontal plane through the baseline for the vertical
location of the center of gravity (VCG), a vertical transverse
plane either through amidships or through the forward
perpendicular for the longitudinal location (LCG), and a
vertical plane through the centerline for the transverse
position (TCG). (The TCG is usually very nearly in the
centerline plane and is often assumed to be in that plane.)

7.  Displacement and Center of Buoyancy
 It has been shown that the force of buoyancy is equal
to the weight of the displaced liquid and that the
resultant of this force acts vertically upward through a
point called the center of buoyancy, which is the CG of
the displaced liquid (centroid of the immersed
volume).
 Application of these principles to a ship, submarine, or
other floating structure makes it possible to evaluate
the effect of the hydrostatic pressure acting on the
hull and appendages by determining the volume of the
ship below the waterline and the centroid of this
volume. The submerged volume, when multiplied by
the specific weight of the water in which the ship
floats is the weight of displaced liquid and is called the
displacement, denoted by the Greek symbol Δ.

8.  The attitude of a floating object is determined by
the interaction of the forces of weight and
buoyancy. If no other forces are acting, it will
settle to such a waterline that the force of
buoyancy equals the weight, and it will rotate
until two conditions are satisfied:
 The centers of buoyancy B and gravity G are in
the same vertical line, as in Fig. 1(a).

Any slight clockwise rotation from this position,
as from WL to W1L1 in Fig. 1(b), will cause the
center of buoyancy to move to the right, and the
equal forces of weight and buoyancy to generate
a couple tending to move the object back to float
on WL (this is the condition of stable
equilibrium).

10.  For every object, with one exception as noted later, at
least one position must exist for which these
conditions are satisfied, since otherwise the object
would continue to rotate indefinitely. There may be
several such positions of equilibrium. The CG may be
either above or below the center of buoyancy, but for
stable equilibrium, the shift of the center of buoyancy
that results from a small rotation must be such that a
positive couple (in a direction opposing the rotation)
results.
 An exception to the second condition exists when the
object is a body of revolution with its CG exactly on
the axis of revolution, as illustrated in Fig. 2. When
such an object is rotated to any angle, no moment is
produced, since the center of buoyancy is always
directly below the CG. It will remain at any angle at
which it is placed (this is a condition of neutral
equilibrium).

12.  A submerged object whose weight equals its
buoyancy that is not in contact with the seafloor
or other objects can come to rest in only one
position. It will rotate until the CG is directly
below the center of buoyancy. If its CG coincides
with its center of buoyancy, as in the case of a
homogeneous object, it would remain in any
position in which it is placed since in this case it
is in neutral equilibrium.
 The difference in the action of floating and
submerged objects is explained by the fact that
the center of buoyancy of the submerged object
is fixed relative to the body, while the center of
buoyancy of a floating object will generally shift
when the object is rotated as a result of the
change in shape of the immersed part of the
body.

13.  As an example, consider a watertight body
having a rectangular section with dimensions
and CG as illustrated in Fig. 3. Assume that it
will float with half its volume submerged, as in
Fig. 4. It can come to rest in either of two
positions, (a) or (c), 180 degrees apart. In
either of these positions, the centers of
buoyancy and gravity are in the same vertical
line. Also, as the body is inclined from (a) to
(b) or from (c) to (d), a moment is developed
which tends to rotate the body back to its
original position, and the same situation
would exist if it were inclined in the opposite
direction.

16.  If the 20-cm dimension were reduced with the CG still
on the centerline and 2.5 cm below the top, a situation
would be reached where the center of buoyancy would
no longer move far enough to be to the right of the CG
as the body is inclined from (a) to (b). Then the body
could come to rest only in position (c).
 As an illustration of a body in the submerged
condition, assume that the weight of the body shown
in Fig. 3 is increased so that the body is submerged, as
in Fig. 5. In positions (a) and (c), the centers of
buoyancy and gravity are in the same vertical line. An
inclination from (a) in either direction would produce a
moment tending to rotate the body away from position
(a), as illustrated in Fig. 5(b). An inclination from (c)
would produce a moment tending to restore the body
to position (c). Therefore, the body can come to rest
only in position (c).

18.  A ship or submarine is designed to float in the
upright position. This fact permits the
definition of two classes of hydrostatic
moments, illustrated in Fig. 6, as follows:
 Righting moments: A righting moment exists
at any angle of inclination where the forces of
weight and buoyancy act to move the ship
toward the upright position.
 Overturning moments: An overturning
moment exists at any angle of inclination
where the forces of weight and buoyancy act
to move the ship away from the upright
position.

20.  The center of buoyancy of a ship or a surfaced submarine moves
with respect to the ship, as the ship is inclined, in a manner that
depends upon the shape of the ship in the vicinity of the
waterline. The center of buoyancy of a submerged submarine, on
the contrary, does not move with respect to the ship, regardless
of the inclination or the shape of the hull, since it is stationary at
the CG of the entire submerged volume. This constitutes an
important difference between floating and submerged ships. The
moment acting on a surface ship can change from a righting
moment to an overturning moment, or vice versa, as the ship is
inclined, but this cannot occur on a submerged submarine unless
there is a shift of the ship's CG.
 It can be seen from Fig. 6 that lowering of the CG along the ship's
centerline increases stability. When a righting moment exists,
lowering the CG along the centerline increases the separation of
the forces of weight and buoyancy and increases the righting
moment. When an overturning moment exists, sufficient lowering
of the CG along the centerline would change the moment to a
righting moment, changing the stability of the initial upright
equilibrium from unstable to stable.

21.  In problems involving longitudinal stability of
undamaged surface ships, we are concerned primarily
with determining the ship's draft and trim under the
influence of various upsetting moments, rather than
evaluating the possibility of the ship capsizing in the
longitudinal direction. If the longitudinal centers of
gravity and buoyancy are not in the same vertical line,
the ship will change trim as discussed in Section 8 and
will come to rest as illustrated in Fig. 7, with the
centers of gravity and buoyancy in the same vertical
line.
 A small longitudinal inclination will cause the center of
buoyancy to move so far in a fore and aft direction
that the moment of weight and buoyancy would be
many times greater than that produced by the same
inclination in the transverse direction. The longitudinal
shift in buoyancy creates such a large longitudinal
righting moment that longitudinal stability is usually
very great compared to transverse stability.

23.  Thus, if the ship's CG were to rise along the
centerline, the ship would capsize transversely long
before there would be any danger of capsizing
longitudinally. However, a surface ship could,
theoretically, be made to founder by a downward
external force applied toward one end, at a point near
the centerline, and at a height near or below the
center of buoyancy without capsizing. It is unlikely,
however, that an intact ship would encounter a force
of the required magnitude.
 Surface ships can, and do, founder after extensive
flooding as a result of damage at one end. The loss of
buoyancy at the damaged end causes the center of
buoyancy to move so far toward the opposite end of
the ship that subsequent submergence of the
damaged end is not adequate to move the center of
buoyancy back to a position in line with the CG, and
the ship founders, or capsizes longitudinally.

24.  In the case of a submerged submarine, the center of
buoyancy does not move as the submarine is inclined
in a fore-and-aft direction. Therefore, capsizing of an
intact submerged submarine in the longitudinal
direction is possible and would require very nearly
the same moment as would be required to capsize it
transversely. If the CG of a submerged submarine
were to rise to a position above the center of
buoyancy, the direction, longitudinal or transverse, in
which it would capsize would depend upon the
movement of liquids or loose objects within the ship.
The foregoing discussion of submerged submarines
does not take into account the stabilizing effect of
the bow and stern planes which have an important
effect on longitudinal stability while the ship is
underway with the planes producing hydrodynamic
lift.

25.  The magnitude of the upsetting forces, or
heeling moments, that may act on a ship
determines the magnitude of moment that must
be generated by the forces of weight and
buoyancy in order to prevent capsizing or
excessive heel.
 External upsetting forces affecting transverse
stability may be caused by:
 Beam winds, with or without rolling.
 Lifting of heavy weights over the side.
 High-speed turns.
 Grounding.
 Strain on mooring lines.
 Towline pull of tugs.

27.  Internal upsetting forces include:
 Shifting of on-board weights athwartship.
 Entrapped water on deck.
 The discussion below is general in nature and illustrates
the stability principles involved when a ship is subjected to
upsetting forces.
 When a ship is exposed to a beam wind, the wind pressure
acts on the portion of the ship above the water-line, and
the resistance of the water to the ship's lateral motion
exerts a force on the opposite side below the waterline.
The situation is illustrated in Fig. 8. Equilibrium with
respect to angle of heel will be reached when:
 The ship is moving to leeward with a speed such that the
water resistance equals the wind pressure, and
 The ship has heeled to an angle such that the moment
produced by the forces of weight and buoyancy equals the
moment developed by the wind pressure and the water
pressure.

28.  As the ship heels from the vertical, the wind pressure, water
pressure, and their vertical separation remain substantially
constant. The ship's weight is constant and acts at a fixed point.
The force of buoyancy also is constant, but the point at which it
acts varies with the angle of heel. Equilibrium will be reached
when sufficient horizontal separation of the centers of gravity and
buoyancy has been produced to cause a balance between heeling
and righting moments.
 When a weight is lifted over the side, as illustrated in Fig. 9, the
force exerted by the weight acts through the outboard end of the
boom, regardless of the angle of heel or the height to which the
load has been lifted. Therefore, the weight of the sidelift may be
considered to be added to the ship at the end of the boom. If the
ship's CG is initially on the ship's centerline, as at G in Fig. 9, the
CG of the combined weight of the ship and the sidelift will be
located along the line GA and will move to a final position, G1
when the load has been lifted clear of the pier. Point G1 will be off
the ship's centerline and somewhat higher than G. The ship will
heel until the center of buoyancy has moved off the ship's
centerline to a position directly below point G1.

30.  Movement of weights already aboard the
ship, such as passengers, liquids, or cargo,
will cause the ship's CG to move. If a weight
is moved from A to B in Fig. 10, the ship's CG
will move from G to G1 in a direction parallel
to the direction of movement of the shifted
weight. The ship will heel until the center of
buoyancy is directly below point G1.

32.  When a ship is executing a turn, the dynamic
loads from the control surfaces and external
pressure accelerate the ship towards the center of
the turn. In a static evaluation, the resulting
inertial force can be treated as a centrifugal force
acting horizontally through the ship's CG. This
force is balanced by a horizontal water pressure
on the side of the ship, as illustrated in Fig. 11(a).
 Except for the point of application of the heeling
force, the situation is similar to that in which the
ship is acted upon by a beam wind, and the ship
will heel until the moment of the ship's weight and
buoyancy equals that of the centrifugal force and
water pressure.

34.  If a ship runs aground in such a manner that contact with
the seafloor occurs over a small area (point contact), the
sea bottom offers little restraint to heeling, as illustrated in
Fig. 11(b), and the reaction between ship and seafloor of
the bottom may produce a heeling moment. As the ship
grounds, part of the energy due to its forward motion may
be absorbed in lifting the ship, in which case a reaction, R,
between the bottom and the ship would develop.
 This reaction may be increased later as the tide ebbs.
Under these conditions, the force of buoyancy would be
less than the weight of the ship because the ship would be
supported by the combination of buoyancy and the
reaction at the point of contact. The ship would heel until
the moment of buoyancy about the point of contact
became equal to the moment of the ship's weight about the
same point, when (W−R) × a equals W × b.

35.  There are numerous other situations in which external
forces can produce heel. A moored ship may be
heeled by the combination of strain on the mooring
lines and pressure produced by wind or current.
Towline strain may produce heeling moments in either
the towed or towing ship. In each ease, equilibrium
would be reached when the center of buoyancy has
moved to a point where heeling and righting moments
are balanced.
 In any of the foregoing examples, it is quite possible
that equilibrium would not be reached before the ship
capsized. It is also possible that equilibrium would
not be reached until the angle of heel became so large
that water would be shipped through topside
openings, and that the weight of this water, running
to the low side of the ship, would contribute to
capsizing which otherwise would not have occurred.

36.  Upsetting forces act to incline a ship in the longitudinal as well as
the transverse direction. Since a surface ship is much stiffer,
however, in the longitudinal direction, many forces, such as wind
pressure or towline strain, would not have any significant effect in
inclining the ship longitudinally. Shifting of weights aboard in a
longitudinal direction can cause large changes in the attitude of
the ship because the weights can be moved much farther than in
the transverse direction. When very heavy lifts are to be
attempted, as in salvage work, they are usually made over the
bow or stern rather than over the side, and large longitudinal
inclinations may be involved in these operations.
 Stranding at the bow or stern can produce substantial changes in
trim. In each ease, the principles are the same as previously
discussed for transverse inclinations. When a weight is shifted
longitudinally or lifted over the bow or stern, the CG of the ship
will move, and the ship will trim until the center of buoyancy is
directly below the new position of the CG. If a ship is grounded at
the bow or stern, it will assume an attitude such that the
moments of weight and buoyancy about the point of contact are
equal.

37.  In the case of a submerged submarine, the center
of buoyancy is fixed, and a given upsetting
moment produces very nearly the same
inclination in the longitudinal direction as it does
in the transverse direction (Fig. 12). The only
difference, which is trivial, is because of the effect
of liquids aboard which may move to a different
extent in the two directions.
 A submerged submarine, however, is
comparatively free from large upsetting forces.
Shifting of the CG as the result of weight changes
is carefully avoided. For example, when a torpedo
is fired, its weight is immediately replaced by an
equal weight of water at the same location.