Metacentric height (GM) is the stability figure most often quoted on the bridge and in the planning office. This guide goes deeper into GM: where KM and KG come from, how the free surface correction works, what happens when cargo is added or shifted, and why GM can be too small as well as too large. The basics are in ship stability basics for planners.
The core formula
GM = KM − KG − FSC
| Component | Set by | How you get it |
|---|---|---|
| KM | Hull form and draft | Hydrostatic tables in the stability booklet, at the current displacement or draft |
| KG | How the ship is loaded | Σ(weight × VCG) / Σ weight, including lightship |
| FSC | Slack tanks | Σ free surface moments / displacement |
KM from hydrostatics. KM = KB + BM, where BM = waterplane moment of inertia / underwater volume. Planners do not calculate it themselves; they read KM from the hydrostatic tables at the right draft. KM changes with draft, so any large change in loading means reading the tables again.
KG from loading. Lightship KG comes from the inclining experiment and is stated in the stability booklet. The rest is worked out from each weight's moment about the keel, exactly like the combined centre of gravity of cargo inside a container.
FSC from tanks. Every tank that is not full raises G virtually. The booklet usually lists the maximum free surface moment for each tank; their sum divided by displacement gives the correction.
Worked example: adding and shifting weight
Illustrative ship, not a real vessel. Initial condition: Δ = 12,000 t, KG = 7.400 m, KM = 8.600 m, total free surface moments 1,800 t·m.
- Solid GM = 8.600 − 7.400 = 1.200 m.
- FSC = 1,800 / 12,000 = 0.150 m, so fluid GM = 1.050 m.
Step 1: load 600 t of containers on deck at VCG 13.0 m.
- New Δ = 12,600 t.
- New KG = (12,000 × 7.400 + 600 × 13.0) / 12,600 = (88,800 + 7,800) / 12,600 = 7.667 m.
- KM at Δ 12,600 t, read again from the hydrostatics: 8.580 m (example).
- FSC = 1,800 / 12,600 = 0.143 m.
- Fluid GM = 8.580 − 7.667 − 0.143 = 0.770 m.
GM has dropped by 0.28 m simply because 600 t went high on deck.
Step 2: shift 200 t from deck (VCG 13.0 m) to the hold (VCG 4.0 m).
- Shift of G = w × d / Δ = 200 × 9.0 / 12,600 = 0.143 m downwards.
- New KG = 7.667 − 0.143 = 7.524 m. Displacement and KM are unchanged.
- Fluid GM = 8.580 − 7.524 − 0.143 = 0.913 m, or 0.91 m rounded.
Step 3: the transverse effect. Shifting 50 t by 10 m to one side moves G sideways by 50 × 10 / 12,600 = 0.040 m. The resulting list ≈ arctan(0.040 / 0.913) ≈ 2.5°. The smaller the GM, the larger the list for the same weight shift.
Repeat these steps with your own numbers in the basic GM calculator and the combined centre of gravity tool.
Too little GM: a tender ship
A ship with small GM is called tender. Signs and risks:
- slow rolling to large angles under beam wind, hard turns or waves;
- a large list from a small weight shift, as in the example above;
- a safety margin wiped out by small errors, such as a wrong VGM or an unrecorded tank;
- at large angles the GZ curve can fail the IS Code criteria even while GM is still positive.
Too much GM: a stiff ship
A ship with large GM is called stiff. It returns upright quickly, but its roll is short and jerky. High transverse accelerations increase the forces on lashing and twistlocks, can collapse deck container stacks, shift breakbulk cargo and make life hard for the crew. Lashing calculations in the Cargo Securing Manual or lashing software take GM as an input, and some ships state a maximum GM. See container lashing on ships, and estimate lashing forces with the lashing calculator.
Rolling period: a rough check
A common rule of thumb (an approximation, not the official formula for your ship):
T ≈ C × B / √GM, where T is the rolling period in seconds, B the ship's breadth in metres and C roughly 0.7 to 0.8.
For an illustrative ship with B = 25 m and C = 0.78:
| Fluid GM (m) | Approximate rolling period (s) | Character |
|---|---|---|
| 0.30 | about 36 | Very tender, take care |
| 0.91 | about 20 | Moderate, usually comfortable |
| 3.00 | about 11 | Stiff, high lashing forces |
These figures are illustrative only. The coefficient C depends on weight distribution and hull form; the IS Code uses a formula of the same shape, with a coefficient derived from the ship's dimensions, for its weather criterion. A rolling period measured at sea is useful for spotting something odd, not for replacing the calculation.
GM changes during the voyage
Departure GM is not the GM for the whole passage. Several things move it at sea:
- Fuel and fresh water consumption. Double bottom tanks sit low. As they are used, low weight disappears, so KG rises and GM falls. Tanks that start to empty also add free surface.
- Ballast operations. Moving or exchanging ballast at sea changes KG and can leave several tanks slack at the same time.
- Deck cargo that absorbs water. Timber or other porous cargo on deck can gain weight from rain and spray, and that weight sits high.
- Intermediate ports. On a multi-port rotation, every discharge and load creates a new condition that has to be checked again.
That is why the arrival condition and the worst intermediate condition must be calculated, not just departure.
Common mistakes when calculating GM
- Using KM for the original draft after hundreds of tonnes have been added.
- Leaving out the FSC, or counting only the free surface of the tanks in use.
- Getting the sign of a weight shift wrong: moving weight up raises KG, moving it down lowers KG.
- Using one container VCG for every tier instead of the actual tier heights, for example high cubes in the upper deck tiers.
- Relying on GM without the GZ curve, even though the IS Code criteria also judge the righting lever at large angles.
Minimum GM from the stability booklet
The stability booklet usually contains a minimum GM (or maximum KG) curve against draft, sometimes against trim too. The curve already reflects the IS Code 2008 intact criteria and, for some ships, other required criteria. To use it:
- Calculate fluid GM for each condition: departure, arrival and the worst intermediate stage.
- Read the minimum GM at that condition's draft and trim, interpolating where needed.
- Make sure fluid GM exceeds the minimum with a sensible margin and stays below any upper limit.
The drafts and trim used to read the curve are explained in ship trim and draft. STOWAGE.ID Vessel Stowage calculates GM and compares it with the minimum GM curve from shore as planning support. The vessel's class-approved loading computer and stability booklet remain the official reference.