STATIC STABILITY
When we say a boat is stable we mean it will (a) float upright when at rest in still water and (b) return to its initial upright position if given a slight, temporary deflection to either side (that is, heeled) by some external force. We then say it has positive stability. An unstable craft, on the other hand, if slightly deflected, will continue to heel and finally come to rest at some other position. In the worst case it will capsize. Such a vessel is said to have negative stability. In the rare case of neutral stability, if heeled to some angle by an external force that is then removed, the vessel will remain at the same angle of heel. The following figure shows how a cone can be stable, unstable, or neutrally stable depending on its initial position.
This lesson will be confined to the special case of ships at rest in still water. Questions of motions resulting from waves are not considered at this stage. How naval architects analyze the question of whether a proposed ship (when floating in still water) is likely to capsize is going to be explained. This requires an understanding of what is called transverse static stability. Also longitudinal static stability (that is, relative to trimming fore and aft) and the special case of submerged submarines will be considered. The subjects of intact, undamaged hulls and then what may happen if the shell is ruptured and flooding results will be touched.
Transverse Static Stability The metacenter: Except in heavily ballasted, deep-keel sailboats, the typical vessel's center of gravity is higher than its center of buoyancy. That is, its center of weight (pushing down) is above its center of support (pushing up). You might expect such a combination to be unstable. That is not necessarily the case, however. In a floating body, an imposed angle of heel causes the center of buoyancy to move sideways from its initial position on the centerline to a new position somewhere in the more deeply immersed side. This is shown by the shift of B to B in the following Figure.
The ship's weight (W) and buoyancy (or displacement) are of course equal, and both continue to act vertically even though the vessel is heeled. Please note that the values of both weight and buoyancy are symbolized by the Greek letter Delta (D). In a stable ship the new center of buoyancy moves far enough to provide a turning moment (or couple) that will try to return the ship to the initial upright position. When that is true, the buoyant line of force will intersect the ship's centerline somewhere above the center of gravity. Naval architects call this intersection the metacenter because it is the position at which the buoyancy seems to be acting. The metacenter's location is shown at point M in the figure, and that is its standard abbreviation.
Buoyant stability is somewhat analogous to forces acting on a rocking chair (see Fig.). Again, the center of weight is higher than the center of support (which is down on the floor). When the person in the chair leans back, his or her weight moves, but the center of support also moves and so keeps the chair from tipping over. If you want to see how raising a weight can lead to instability, try standing up in a rocking chair-or a canoe (better wear a bathing suit for this latter experiment).
Metacentric height Referring again to the figure, suppose something were to raise the center of gravity from where it is shown (at G) to some location above the metacenter, M. Then, obviously, the turning moment would tend to increase the angle of heel and the vessel would be in a condition of negative stability. The distance between G and M is called the metacentric height and is considered positive if G is below M, or negative if M is below G. In plain words, positive stability is found whenever the position of the metacenter is above the vertical center of gravity, and negative stability arises when the positions are reversed.
The metacentric height is usually referred to simply as "GM" It is the key indicator of initial transverse stability. The angle of heel is usually referred to by the Greek letter theta (q). For ordinary hull forms, GM remains essentially constant for small values of (q), say up to 7 degrees. If greater angles are imposed, GM will at first tend to increase, but will then decrease and eventually become neutral, then negative, and a capsize will result.
Righting arm The distance GZ shown in Figure is called the righting arm (or righting lever), and the righting moment equals D times GZ. At small angles of heel, GZ equals GM times the sine of the angle q, that is, GZ = GM sinq
Predicting metacentric height Clearly, the naval architect must know how to find GM. This is a three-step process. The first step is to find KB, the height of the center of buoyancy above the baseline. An exact value can be derived from numerical analysis. For ordinary ship forms, KB will come out to be about 52 percent of the draft. Usually, fine ship hull lines (that is, the ship having a low block coefficient) will tend to raise KB, as will also a V-bottom hull form.
The next step is to find the distance BM, which naval architects call the metacentric radius. It can be shown that Where I is the transverse moment of inertia of the ship' s waterplane about its own centerline, and V is the volume of displacement. Again, an exact value of I can be found by numerical analysis. In the early stages of design, and before the lines are drawn, approximate values are found recognizing that I will be directly proportional to the ship's length multiplied by the cube of the beam. From this you can infer the importance of a wide beam in providing stability to your boat.
The naval architect's third task is to find KG, the vertical center of gravity of all weights: the empty ship plus all the deadweight items. Obviously, one never knows exactly how a ship will be loaded, so the figure for KG can never be more than an estimate. The prudent naval architect will assume it is rather high. The naval architect now has all the ingredients needed to predict the metacentric height, GM. Referring to figure you can see that KM=KB+BM GM=KM-KG You may ask: Will stability be affected by moving from salt water to fresh? The lesser density of fresh water will allow the vessel to sink to a slightly greater draft, but the net overall effect on stability will be insignificant.
The inclining experiment In an existing vessel, GM can be found by means of what naval architects call the inclining experiment. In this exercise a known weight W is moved from the vessel's centerline to a distance (d) to one side (see Figure). The resulting angle of heel, q, produces a lateral shift of the pendulum. How far will the vessel heel? It will heel until the righting moment (RM) exactly balances the heeling moment (HM).This leads to the simple expression for the metacentric height: RM=HM D. GZ = W. d GZ = GM sinq
In practice naval architects substitute tanq for sinq. They do so because tan q is more easily measured (as the distance the pendulum swings divided by the length of the pendulum) and sine and tangent are virtually equal at small angles. This leads to: It is a very simple experiment, you can try this experiment on a small boat. But be careful. Do it on a calm day. Make sure the mooring lines are slack. Get rid of all bilge water and other loose liquids, and make everyone on board hold perfectly still.
Ideal degree of stability In designing a ship, the naval architect aims for a reasonable value of GM. Values of perhaps three to five percent of the beam are usually considered about right (much less than shown in Figure). Smaller values leave too little margin in case of accident, careless loading, topside icing, or whatever. Greater values on the other hand, lead to excessive stability. Remember that when we say a ship is stable we usually mean it is stable with respect to the water's surface. Thus, a ship with great stability will automatically follow every wave profile that comes along. That will lead to short, uncomfortable rolling that may be dangerous to life, limb, and digestive system alike. Harsh rolling may also produce excessive stresses on rigging, stacks, and other elevated objects.
Free Surface Effects An unpleasant surprise that can ruin stability may result from internal liquids of any sort. Assume such liquids are not snugly contained. If the ship heels to one side the liquids will flow in the same direction, thus shifting the ship's overall center of gravity toward the low side. This has the effect of reducing the metacentric height. Of course, if the internal liquid completely fills its tank, there will be no free-surface effect on stability. Loose bulk cargos such as grain can easily shift, so their free surfaces must somehow be controlled if stability is to be maintained. In tankers, longitudinal bulkheads (usually two of them) are used to reduce the width of the cargo's free surface (they also reduce the danger of excessive dynamic forces resulting from liquids sloshing about).