Flotation, Centres and Metacentric Height

How a ship floats: mass, volume and density

In short: This is the first Stability topic: start here. Every stability sum uses three quantities — mass, volume and density — and one ratio, relative density (RD).

• Mass: how much matter there is. On board we work in tonnes (t).
• Volume: the space something takes up, in cubic metres (m³).
• Density: mass per unit volume. Density = mass ÷ volume, in t/m³. So mass = volume × density.
• Typical values: fresh water 1.000 t/m³, sea water 1.025 t/m³. Dock and river water lies in between.
• Relative density (RD): the density of a substance ÷ the density of fresh water. It has no units. Sea water: RD 1.025. An oil with RD 0.85 floats on water.
• On the bridge: a tank of 50 m³ filled with sea water holds 50 × 1.025 = 51.25 t; the same tank filled with fuel of RD 0.85 holds 42.5 t.

Archimedes and the law of flotation

In short: A body in a fluid is pushed up by the weight of the fluid it displaces (Archimedes). A floating ship sinks until she displaces her own weight of water (law of flotation).

• Archimedes’ principle: a body wholly or partly immersed in a fluid feels an upthrust equal to the weight of fluid displaced.
• Law of flotation: a floating body displaces a weight of liquid equal to its own weight. That weight is the ship’s displacement (Δ).
• Displacement = underwater volume × density of the water.
• Weight acts down through G, buoyancy acts up through B. Floating upright and at rest, the two forces are equal and on the same vertical line.
• On the bridge: going from the sea into a fresh-water river, the weight does not change but the water is less dense. She must displace a bigger volume, so she sinks deeper (bigger draught).

Draught and form coefficients

In short: The draught is how deep the keel is below the waterline. Form coefficients compare the real hull with a box of the same length, breadth and draught: they describe how “fine” or “full” she is.

• Draught: vertical distance from the keel to the waterline, read on the draught marks forward, aft and (on bigger ships) amidships.
• Waterplane area coefficient (Cw) = waterplane area ÷ (L × B).
• Block coefficient (Cb) = underwater volume ÷ (L × B × draught).
• Midship section area coefficient (Cm) = immersed midship area ÷ (B × draught).
• All three are less than 1. A box barge is close to 1; a fast, fine yacht hull is much lower.
• They matter because they link the draught to volume and area: the underwater volume gives the displacement, the waterplane area gives TPC (tonnes per centimetre immersion).

The four points of stability: K, B, G and M

In short: Four points on the centreline do all the work: K (keel), B (centre of buoyancy), G (centre of gravity) and M (metacentre). All heights are measured up from K.

K — the keel

• The reference point at the bottom. KB, KG and KM are heights above the keel.

B — centre of buoyancy

• The centre of the underwater volume (the geometric centre of the immersed hull). Buoyancy acts up through B.
• B moves whenever the underwater shape changes: it rises as the draught increases and moves towards the low side when she heels.

G — centre of gravity

• The point through which the whole weight of the ship acts downwards.
• G moves only when weight is added, removed or shifted on board. Heeling alone does not move G.
• The lightship weight and KG come from the inclining experiment, done in lightship condition (REG YC 11.9(1)). For each loading condition, KG is then calculated by moments (see below).

M — the metacentre

• For small angles of heel (textbooks usually say up to about 10–15°), M is the point where the vertical through the new B cuts the centreline.
• In this range M stays practically fixed, so GM is a good measure of initial stability.
• KM = KB + BM. BM (the metacentric radius) depends on the shape of the waterplane, so KM changes with draught.

GM = KM − KG and the hydrostatic tables

In short: GM = KM − KG. KM comes from the hydrostatic tables in the stability booklet at the present draught; KG comes from your own calculation. GM is always calculated — no table gives it to you.

• KM: enter the hydrostatic tables (part of the approved Stability Information Booklet, REG YC 11.10) with the mean draught or displacement and read KM.
• KG: start from lightship weight and KG, add every weight on board (fuel, water, stores, tenders, people) with its own height. KG = sum of (weight × height) ÷ total weight.
• GM = KM − KG, then correct for free surface to get the effective (fluid) GM.
• Example: KM 6.20 m, KG 5.60 m → GM 0.60 m (before free surface correction).
• Hydrostatic tables give, for each draught: displacement, TPC, KMT (transverse KM), KML (longitudinal KM), KB and LCB (longitudinal centre of buoyancy). They describe the hull shape, so they never give GM: GM depends on how you have loaded her.
• For large commercial yachts, the GM after free surface correction must be at least 0.15 m (REG YC 11.2(1)(a)(v)).

Three states of equilibrium

In short: Heel her a little and see what G and B do. G below M → stable (GM positive). G at M → neutral (GM = 0). G above M → unstable (GM negative).

• Stable (positive GM): G is below M. When she heels, B moves to the low side and the buoyancy pushes up outboard of G. The righting couple brings her back upright.
• Neutral (GM = 0): G is at M. Weight and buoyancy act on the same vertical line at any small heel: no couple, so she stays at whatever angle she is pushed to.
• Unstable (negative GM): G is above M. The couple acts the wrong way and heels her further.
• GZ (righting lever) = the horizontal distance between G and the vertical through B. Righting moment = Δ × GZ. At small angles GZ ≈ GM × sin θ.
• The bigger the GM, the more stable she is at small angles. The bigger the GZ, the more force she has to come back upright.
• But more is not always better: a very large GM gives a short, jerky roll that is hard on people, gear and lashings.

Adding and removing weight

In short: G moves towards a weight you add and away from a weight you remove — along the line joining G and that weight’s centre of gravity.

• Load a weight → G moves towards it. Discharge a weight → G moves away from it. Shift a weight on board → G moves parallel to the shift.
• Size of the shift: GG1 = (w × d) ÷ (Δ + w) when loading, ÷ (Δ − w) when discharging (w = weight, d = distance from G to the weight).
• Weight added high (above G) raises G and reduces GM. Weight added low (double-bottom tanks) lowers G and increases GM. Removing weight from low down (using fuel from double bottoms) raises G.
• Adding or removing weight also changes the draught, so KM changes too: always re-read KM before working out the new GM.
• A weight lifted by a crane acts at the crane head as soon as it leaves the deck: G jumps up at once.
• On the bridge: Δ 1,000 t, you lift a 20 t tender onto a deck 4 m above G. GG1 = 20 × 4 ÷ 1,020 = 0.08 m: G rises 0.08 m and GM falls by about the same.
• The same logic explains why you ballast low tanks before heavy weather, why fire-fighting water left on board is a danger when fighting a fire, and why flooding after a collision or grounding can capsize a ship.