Intact Stability: Elementary Principles

By Mazlan Muslim, MEng, UniKL MIMET

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.

Concepts of Equilibrium

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.

Neutral & Unstable Equilibrium

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.

Weight and Center of Gravity

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.

The position of the CG

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.)

Location of the CG

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.

Displacement and Center of Buoyancy

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).

Interaction of Weight and Buoyancy

Figure 1: Stable equilibrium of floating body

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).

Shift of the center of buoyancy

Figure 2: Neutral equilibrium of floating body

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.

Submerged object

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.

Example

Figure 3: Example of stability of watertight rectangular body

Figure 4: Alternate conditions of stable equilibrium for floating body

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).

Body in the submerged condition

Figure 5: Single condition of stable equilibrium for submerged floating body

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.

Righting moments

Figure 6: Effect of height of CG on stability

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.

Floating and submerged ships

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.

Longitudinal stability

Figure 7: Longitudinal equilibrium

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.

Extensive flooding

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.

Submerged submarine

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.

Upsetting Force

Figure 8: Effect of a beam wind

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.

Internal upsetting forces

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.

Force of buoyancy

Figure 9: Lifting a weight over the side

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.

Movement of weights

Figure 10: Effect of offside weight

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.

Heeling force

Figure 11: Effect of a turn and grounding

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.

Ship grounding

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.

Ship capsizing

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.

Upsetting forces

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.

Shifting of the CG

Figure 12: Effect of weight shift on the transverse and longitudinal stability of a submerged submarine