Stability

Heel a ship and two forces stay in play: its weight acting down through G, and its buoyancy acting up through B. Upright, they are in the same vertical line and cancel. Heeled, the underwater shape is no longer symmetrical — more volume on the low side, less on the high side — so B moves towards the low side while G stays put.

The two forces are now offset. They form a couple, and the horizontal distance between their lines of action is the righting lever GZ.

That is the whole of stability. Everything else is a way of finding GZ at a given angle, or of noticing something that has made it smaller.

For small angles there is a shortcut. B moves along a circular arc whose centre is the metacentre M, and the geometry gives GZ ≈ GM·sin θ. This holds while the arc assumption holds — roughly to 10°, sometimes to 15° on a full hull. Beyond that the deck edge approaches the water, the underwater shape changes character, and the shortcut stops being true.

Four heights, all measured from the keel, and the distinction between them is examinable in its own right.

• KB — height of the centre of buoyancy. Depends on hull shape and draught. Roughly 0.53 T on a normal merchant hull.
• BM — metacentric radius, equal to I_T/∇. Depends on the waterplane and the displaced volume.
• KM = KB + BM. Still pure geometry: it is a property of the ship at a given draught, printed in the hydrostatic tables, and no amount of loading changes it.
• KG — height of the centre of gravity. This is the loading, and it is the only one the ship's staff can change.

Because BM = I/∇ and I depends on breadth cubed, beam is the strongest lever on initial stability. Doubling the beam of a rectangular waterplane multiplies I by eight while multiplying ∇ by only two, so BM rises by a factor of four.

KG is found by taking moments of every weight on board about the keel, and its accuracy limits everything downstream. The inclining experiment is how the lightship KG is established in practice: known weights are shifted across the deck and the resulting small angle is measured with a pendulum.

The experiment is conducted with the ship as nearly empty as possible, tanks pressed up or dry, no free surface, no mooring restraint and calm conditions — every one of those precautions exists to remove something that would corrupt the measured angle.

A partly filled tank is dangerous in a way that a full tank is not. Heel the ship and the liquid runs to the low side. Its own centre of gravity shifts, and the ship's effective centre of gravity shifts with it — reducing GZ at every angle.

The convenient way to account for it is as a virtual rise of G: pretend G has moved up by an amount that produces the same loss.

Three consequences follow, and each one has been an examination question:

• Breadth is everything. i depends on b³. A tank split down the centreline has each half at half the breadth, so each contributes one eighth — two of them give one quarter of the undivided value.
• The quantity of liquid does not appear. A tank with 10 cm of water gives the same free surface as one half full. This is why an almost-empty tank is not almost-safe.
• The position of the tank does not appear either. High or low, forward or aft, the correction is the same — only the surface dimensions matter.

Always apply the correction to KG, not to KM: the effective GM is KM − (KG + FSC).

Plot GZ against heel angle and the curve reports on the whole range of stability, not just the beginning.

• The initial slope equals GM in radians. Draw a tangent at the origin and read it at 1 radian (57.3°) — a standard construction, and a standard question.
• The maximum is where the righting moment is greatest. Its angle matters: a maximum at 25° or beyond means the ship keeps working for you well past the point a stiff, shallow-curved hull has given up.
• The range of stability is the angle at which GZ returns to zero — beyond it, the ship capsizes.
• The area under the curve is energy: the work needed to heel the ship there. This is why it is called dynamical stability and why gusts and waves, which deliver energy rather than a steady moment, are assessed against area rather than against GZ.

The intact stability criteria are written almost entirely in terms of that area, in metre-radians:

Both leave the ship leaning. The causes are opposite, and so are the remedies — which is exactly why the distinction is examined.

A list is caused by an off-centre weight while GM is positive. The ship settles where the righting moment balances the heeling moment, and it will sit at the same angle steadily.

A loll is caused by negative GM. Upright, the ship has no righting lever at all and will not stay there. It heels until the increasing BM at larger angles has restored GZ to zero, and settles at that angle — on either side, and it may flop from one to the other.

The practical difference: for a list you may correct by moving weight to the high side. Do that to a ship that is lolling and you can capsize her, because the righting lever on the high side is still negative near upright. The correct action for a loll is to lower G first — press up slack tanks one at a time, starting with the low side, and only then think about transverse weight.

The examinable core is the lost buoyancy method. When a compartment floods it stops providing buoyancy; the intact part of the hull must therefore provide more, so the ship sinks and trims until the intact volume has grown by the volume of buoyancy lost.

Permeability μ is the fraction of a space that can actually fill with water — the rest is occupied by structure, machinery or cargo.

Here v is the volume of the flooded space up to the waterline and a is its waterplane area, subtracted because that part of the waterplane no longer contributes to supporting the ship.

Two further ideas complete the outline. Floodable length is the greatest length of compartment that can be flooded without the ship sinking below the margin line, and it is what sets bulkhead spacing. And the modern regulatory approach is probabilistic: rather than requiring survival of one specified damage, it compares an attained subdivision index A — a weighted sum of survival probabilities over many possible damages — against a required index R.

Three problems that between them cover most of what GATE asks of this topic.