Transverse Stability at Small Angles of Heel

CHAPTER 6.

TRANSVERSE STABILITY AT SMALL ANGLES OF HEEL

In this chapter we shall consider the problem of initial stability, which correspond to very small departures from an assumed equilibrium condition. Of particular concern to the ship designer and operator is the transverse stability and consequently we apply a small angle of heel to the ship.

A rigid body floating freely on the surface of a fluid has six degrees of freedom, three of translation and three of rotation. For disturbances from a state of equilibrium, the naval architect refers to the movements in the six degrees as shown in Table 6.1.

Table 6.1.

Movement / Dynamic / Quasi-static
Fore and aft translation / surge / -
Transverse translation / sway / -
Vertical translation / heave / -
Rotation about a fore and aft axis / roll / heel
Rotation about a transverse axis / pitch / trim
Rotation about a vertical axis / yaw / -

6.1. Heeling Forces and Moments

The magnitude of heeling forces and moments determines the forces and moments that must be generated by the forces of weight and buoyancy in order to prevent capsizing or excessive heel.

External heeling forces affecting transvers stability may be caused by;

• Beam winds and waves
• Lifting of heavy weights over the side
• High speed turns
• Grounding
• Strain on mooring lines
• Towline pull of tugs

Internal heeling forces include

• Shifting of on-board weights athwartship
• Entrapped water on deck

Beam winds and waves : When a ship is exposed to a beam wind, the wind pressure acts on the portion of the ship above the waterline, 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 shown in Figure 6.1.

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

Figure 6.1. Effect of beam winds

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 is also constant, but the point 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.

Lifting of Heavy Weights over the Side : When a weight is lifted over the side, as shown in Figure 6.2, 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 en of the boom. If the ship’s centre of gravity is initially on the ship’s centreline, as at G in Figure 6.2, the centre of gravity 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 centreline and somewhat higher than G. The ship will heel until the centre of buoyancy has moved off the ship’s centreline to a position directly below point G1.

High Speed Turns : When a ship is executing a turn, a centrifugal force is generated, which acts horizontally through the ship’s centre of gravity. This force is balanced by a horizontal water pressure on the side of the ship, as shown in Figure 6.3. 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.

Figure 6.2. Lifting of Heavy Weights over the Side

Figure 6.3.High Speed Turns

Grounding : If a ship runs aground in such a manner that the bottom offers little restraint to heeling, as shown in Figure 6.4, the reaction 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 will develop. The force of buoyancy will be less than the weight of the ship, since the ship is supported by the combination of buoyancy and the reaction of the bottom. The ship will heel until until the moment of buoyancy about the point of contact with the bottom becomes equal to the moment of the ship’s weight about the same point, i.e.

(W-R).a = W.b

Figure 6.4. 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 vessel. In each case, equilibrium would be reached when the centre of buoyancy has moved to a point where heeling and righting moments are balanced.

Movement of weight already aboard the ship, such as passengers, liquids or cargo, will cause the ship’s centre of gravity to move. If a weight is moved from A to B, as shown in Figure 6.5, the ship’s centre of gravity will move from G to G1 in a direction parallel to the direction of movement of the shifred weight. The ship will heel until the centre of buoyancy is directly below point G1.

Figure 6.5. Effect of weight movement

6.2. The Transverse Metacentre

Consider a symmetric ship heeled to a very small angle, , as shown in Figure 6.6. For small angles the upright and inclined waterlines will intersect on the waterline. The volumes of the emerged and immersed wedges must be equal for constant displacement. The centre of buoyancy has moved off the ship’s centreline as a result of the inclination, and the lines along which the resultants of weight and buoyancy act are separated by a distance, GZ, the righting arm. A vertical line through the centre of buoyancy will intersect the centreline at a point M, called the transverse metacentre, when .

Figure 6.6. Transverse metacentre and righting arm

Unless there is an abrupt change in the shape of the ship in the vicinity of the waterline, point M will remain practically stationary with respect to the ship as the ship is inclined to small angles, upto about 7-10 degrees.

If the locations of G and M are known, the righting arm for small angles of heel can be calculated readily, with sufficient accuracy, by the formula

The distance GM is therefore important as an index of transverse stability at small angles of heel, and is called the transverse metacentric height. Since GZ is considered positive when the moment of weight and buoyancy tends to rotate the ship toward the upright position, GM is positive when M is above G, and negative when M is below G.

6.2.1. Metacentric Radius (BM)

Consider a symmetric ship heeled to a small angle (), say 2 or 3 degrees, as shown in Figure 6.7.

Figure 6.7. Metacentric radius

For small angles the emerged and immersed wedges are approximately triangular. If y is the half ordinate of the original waterline at the cross section the emerged or immersed section area is.

for a small length dx, the volume of each wedge is

The righting moment is equal to transverse shift of buoyancy.

The total righting moment is

The expression within the integral sign,, is the second moment of area, or the moment of inertia, of a waterplane about its centreline. Hence the movement of buoyancy is

Referring to Figure 6.7, for small angles of heel

Thus the height of the metacentre above the centre of buoyancy is found by dividing the second moment of area of the waterplane about its centreline by the volume of displacement.

6.2.2. Metacentric Diagram

The metacentre diagram is a convenient way of defining variations in relative heights of the centres of buoyancy and metacentre for a series of waterlines parallel to the design waterline. The position of the centres of buoyancy, B, and metacentre, M, are dependent only on the geometry of the ship and the waterplane at which it is floating.

A typical metacentric diagram is shown in Figure 6.8. The vertical scale represents draught and a line is drawn at 45 degrees to this scale. For a given draught, T, a horizontal line is drawn intersecting the 45 degrees line in D and a vertical line is drawn through D. On this vertical line, the distance DM represents the height of the centre of metacentre above the waterplane ans DB represents the depth of the centre of buoyancy below the waterplane. This process is repeated a sufficient number of times to define the loci of metacentre and centre of buoyancy. These are termed the metacentre, M, and buoyancy, B, curves.

A table may be constructed to the left of the diagram, in which are listed the displacements and T1 values for each a number of draughts corresponding to typical ship loading conditions.

Since KB is approximately proportional to draught over the normal operating range, the B curve is usually nearly staright for conventional ship forms. The M curve, on the other hand, usually falls steeply with increasing draught at shallow draught than levels out and may even begin to rise at very deep draught.

Figure 6.8. Metacentric Diagram

6.2.2.1. Metacentric Diagrams for Simple Geometrical Forms

a)Rectangular section

Consider a barge of length L and breadth B with constant rectangular cross sections floating at draught T. The volume of the barge is

The height of the centre of buoyancy is

The metacentric radius is

Then the height of metacentre above the keel is

As can be seen the height of metacentre depends upon the beam and draught but not the length. At zero draught KM would be infinite, and the second term predominates for small draught values.

The draught at which KM is minimum can be found by diffrentiating the equation for KM with respect to T and equating to zero. Then the draught at which KM is minimum can be found as follows,

b)Triangular section

Consider a barge of length L and waterline breadth B with constant triangular cross sections floating apex down at draught T. The volume of the barge is

The height of the centre of buoyancy is

The metacentric radius is

Then the height of metacentre above the keel is

The draught at which KM is minimum can be found by differentiating the equation for KM with respect to T and equating to zero. Then the draught at which KM is minimum can be found as follows,

c)Circular section

Consider a circular cylinder of radius r and centre of section O, floating with its axis horizontal. For any waterline, above or below the centre, and for any inclination, the buoyancy force always acts through the centre. That is, KM is independent of draught and equal to r.

6.3. Measurement of Initial Stability

The difference between the centre of gravity and the metacentre is defined as the metacentric height (GM) and this distance can be used the initial stability of a ship at smalll angles of heel, from 00 to 70-100. For small angles of heel the metacentric height (GM) is calculated by subtracting the height of the ship’s centre of gravity above the keel (KG) from the height of metacentre above the keel (KM), i.e

The height of metacentre above the keel is calculated by the summation of the height of centre of buoyancy above the keel (KB) and the metacentric radius (BM) as follows

where the metacentric radius is the distance between the centre of buoyancy and the metacentre. Therefore

It is evident that GM is the key indicator of initial transverse stability. Whilst it should obviously be positive, too high values should be avoided. GM is a measure of the ship’s stiffness in roll motion and largely governs the period of roll motion. Too high a value of GM leads to a very short roll period.

The actual value of GM for a ship may be found by an inclining experiment, which is discussed in later sections. At the initial stages of design, to ensure that the ship has sufficient initial stability, the metacentric height can be calculated by using approximate formulae.

Posdunine

Morrish

Normand

Schneekluth

Bauer

Henschke

Robb

Riddlesworth

Eames

D’arcengelo

Kiss

Brown

Mc Cloghrie

Posdunine

Rauert

KG=aD

Ship type / a
Tanker / 0.69
Bulk carrier / 0.68
Dry Cargo / 0.72
Passenger / 0.75

6.4. Wall sided formula

For small angles of heel, we found that the righting lever . A more accurate formula for GZ at angles around 100 is available for wall sided ships, i.e. those having vertical sides in the region of the waterline. The vessel can have a turn of bilge provided it is not exposed by the inclination of the ship. Nor must the deck edge be immersed. Because the vessel is wall sided the emerged and immersed wedges will have sections which are right angled triangels of equal area. Consider the wall sided section, shown in Figure 6.9. The transverse moment of volume shift can be expressed as

where y is the half ordinate an I is the second moment of area of the waterplane about the centreline. Therefore

Similarly the vertical moment

Therefore

From Figure 6.9,

Therefore, the righting arm, GZ, is

At small angles, this degenerates to . The formula is invalid at large heel angles.

Figure 6.9.

6.5. Angle of Loll

In certain conditions of loading it is possible for a ship to have negative GM when upright. The GZ curve will have a negative slope at the origin, as shown in Figure 6.10. As the ship heels to larger angles, GZ increase to become positive at an angle, known as the angle of loll. The ship is in unstable equilibrium when upright and will flop over rapidly to the angle of loll.

The wall sided formula can be used to estimate the angle of loll. If the ship has a positive GM it will be in equilibrium when GZ is zero, that is

This condition is satisfied by two values of . The first is , which is the case with the ship upright. The second value is given by

With both GM and BM positive there is no solution to this meaning that the upright position is the only one of equilibrium. This also applies to the case of GM=0, which means that in the upright position the ship is stable, not neutral. When, however, the ship has a negative GM there are two possible solutions for in addition to that of zero, which in this case would be a position of unstable equilibrium. These other solutions are at either side of the upright position being given by,

Such an angle is known as the angle of loll. The ship would show no preference for one side or the other.

Figure 6.10. Angle of loll

If is the angle of loll, the value of GM for small inclinations about the loll position, will be given by the slope of the GZ curve at that point, i.e.

It is important to recognize the loll condition as it is potentially dangerous. It cannot, unlike heel, be corrected by applying a counter moment to the ship. This would merely cause the ship to flop over rapidly to a larger angle on the other side.

6.6. Statical Stability Curve

A typical curve of GZ variation with heel angle is shown in Figure 6.11. This is known as a curve of static stability or GZ curve. Important features of this curve are as follows;

• At small angles . The slope of the GZ curve near the origin is given by . Thus, if the slope of the curve at the origin is extrapolated to a value of radian the ordinate has a value equal to GM, the metacentric height.
• As increases beyond small values, the slope of the GZ curve usually increases above the initial slope. For a wall sided ship, this is predicted by the wall sided formula for GZ (BM is always positive). For a round bilge vessel, this increase in slope may be small or even negative.
• At point B, there is a point of inflexion in the GZ curve and the slope begins to decrease. This change is associated with immersion of the deck edge or emergence of the turn or bilge. At this point, the waterplane width ceases to increase and begins to reduce.
• At point C, GZ reaches its maximum value GZmax. A steady overturning moment applied to the ship of value greater than would cause it to capsize.
• At point D, GZ becomes zero. This point is called the point of vanishing stability and distance OD is called the range of stability.
• The area under the GZ curve is a measure of the work done in steady conditions to heel the ship. It may also be called the dynamical stability, as it is related to the energy absorbing capability of the ship in roll.

GZ curves are normally prodyced for a range of loading conditions. GZ curve is normally corrected for any expected free surface effects in the condition concerned.

There are some deficiencies in GZ curves. For example

• The GZ curve makes no allowance for changes in centre of gravity at very large heel angles, due to cargo shift, for example
• It makes no allowance for water flooding into the ship at large angles, such as through deck openings or engine intakes
• The GZ curve assumes quasi-static conditions which are certainly not present in a heavy sea