Introduction
The metacentric height GM is the first number any officer checks before moving weight on a ship: it says whether the ship will return upright after a small heel and how fast she will roll. Slack tanks reduce it through the free-surface effect, loading high weights reduces it, and any weight off the centreline produces a list that GM alone determines at small angles. This calculator does that initial-stability arithmetic: solid GM from KM and KG, the free-surface correction from the tank tables or a single rectangular tank, the final GM and list after shifting, loading or discharging a weight, the rolling period from the IMO weather-criterion formula, and the IMO minimum-GM check.
It is for deck officers planning cargo and ballast operations, surveyors checking a loading condition, and cadets working the classic stability problems. It covers initial stability only; large-angle stability needs the GZ curve.
Formulas
Metacentric height and free surface:
GM_solid = KM − KG
FSM = Σ (l b³ / 12) × ρ_liquid / n² free surface moment of a rectangular slack tank, t·m
FSC = FSM / Δ free surface correction (virtual rise of G), m
GM_fluid = GM_solid − FSCWeight operations (w tonnes, Kg its height above keel, d its transverse distance from the centreline, + starboard):
Shift: KG′ = KG + w × d_vertical / Δ, heeling moment = w × d, Δ′ = Δ
Load: KG′ = (Δ KG + w Kg) / (Δ + w), heeling moment = w × d, Δ′ = Δ + w
Discharge: KG′ = (Δ KG − w Kg) / (Δ − w), heeling moment = −w × d, Δ′ = Δ − w
GM′_fluid = KM − KG′ − FSM / Δ′List:
tan θ = heeling moment / (Δ′ × GM′_fluid)Rolling period (IMO 2008 IS Code weather criterion):
C = 0.373 + 0.023 (B/d) − 0.043 (L_wl / 100)
T = 2 C B / √GM_fluidIMO minimum: GM_fluid ≥ 0.15 m (2008 IS Code Part A 2.2.4).
- KM — height of the transverse metacentre above keel at the present draft (hydrostatic tables)
- KG — height of the centre of gravity above keel (loading computer / stability booklet)
- l, b — length and breadth of the free liquid surface; n — number of equal longitudinal subdivisions
Assumptions and limitations
- Small-angle (initial) stability: valid for list angles up to about 10°. The calculator warns beyond that; use the GZ curve from the cross curves for the real angle.
- KM is held constant through loading and discharge. For weights above about 2 % of displacement re-enter KM at the new draft.
- Free surface moments are for rectangular tank surfaces (or taken from the tank tables as a total); they are independent of the liquid depth as long as the tank is slack, and they vanish when the tank is pressed up or empty.
- No wind, grain or icing heeling moments; no dynamic effects.
- The rolling period is an IMO estimate for conventional hull forms; the actual period depends on the radius of gyration of the loaded ship.
- Only the 0.15 m GM criterion is checked; the IS Code also requires minimum areas under the GZ curve, a minimum GZ of 0.20 m at 30° and the weather criterion.
Worked example
Bulk carrier: Δ = 47 232 t, KM = 12.50 m, KG = 11.00 m, total free surface moment 1 500 t·m. A 200 t weight is shifted 10 m to starboard on the same deck.
- Solid GM: 12.50 − 11.00 = 1.50 m.
- Free surface correction: 1 500 / 47 232 = 0.032 m → fluid GM 1.468 m.
- Heeling moment: 200 × 10 = 2 000 t·m.
- List: tan θ = 2 000 / (47 232 × 1.468) = 0.0288 → θ = 1.65° to starboard.
- Rolling period: C = 0.373 + 0.023 × 3.2 − 0.043 × 1.8 = 0.369; T = 2 × 0.369 × 32 / √1.468 = 19.5 s — a moderate roll.
- IMO: 1.468 m ≥ 0.15 m, acceptable.
If instead 500 t is loaded at Kg 15 m, 8 m to starboard: Δ′ = 47 732 t, KG′ = (47 232 × 11 + 500 × 15) / 47 732 = 11.042 m, GM′_fluid = 12.50 − 11.042 − 1 500 / 47 732 = 1.427 m, list = atan(4 000 / (47 732 × 1.427)) = 3.36° to starboard.
FAQ
Why does a slack tank reduce GM when the liquid weight has not moved?
As the ship heels the liquid flows to the low side, moving its centre of gravity and reducing the righting moment exactly as if G had risen by FSM/Δ. The effect depends on the breadth cubed, which is why longitudinal subdivision of wide tanks matters so much.
Does it matter how full the slack tank is?
Not for the free-surface moment itself (as long as the surface is free), only for the liquid's own weight and KG. Pressing up or emptying removes the effect entirely.
What GM should a ship have?
At least 0.15 m by the IS Code, but practical values are 0.5–3 m depending on the ship; too much GM gives a stiff, violent roll that damages cargo and fatigues the crew, too little gives a tender ship with a long roll and small margins.
My list came out negative. What does that mean?
A list to port. Transverse distances are positive to starboard, so discharging a weight from the starboard side (negative heeling moment) lists the ship to port.
Can I use this to correct a list?
Yes: the heeling moment needed to bring the ship upright equals the existing one with the opposite sign; divide by the available transverse distance to get the weight to transfer, remembering that the free surface of the ballast being moved also changes GM.
What is the difference between list and loll?
A list is caused by an off-centre weight with positive GM. Loll is the angle a ship with negative GM takes up until the GZ becomes positive again; it is corrected by lowering G, not by transferring weight, and the calculator refuses to compute a list when GM is negative.
Why does the rolling period need L, B and d?
The IS Code formula estimates the radius of gyration from the hull proportions through the coefficient C. A simpler rule, T ≈ 0.8 B / √GM, gives similar numbers.
Related calculators
- Hydrostatics basics — displacement and KM-related hydrostatics at the draft.
- Draft survey — the displacement of the actual condition.
- Bunker quantity and density correction — fuel masses that change KG and free surface.
- Ship squat and under-keel clearance — the other draft-related check before sailing.