⛴️ Calculators

Ship Roll Period (IMO IS Code)

Computes the natural roll period of a ship using the empirical formula of the IMO International Code on Intact Stability (IS Code 2008), used in the weather criterion and in the inclining experiment. The period is T = 2 × C × beam ÷ square root of GM, with the coefficient C = 0.373 + 0.023 × (beam ÷ draught) − 0.043 × (length ÷ 100), so that rolling gets faster as the metacentric height grows. The number indicates comfort and safety: short periods, below some 8 seconds, reveal a stiff ship that rolls with high acceleration and punishes cargo and crew; long periods indicate a tender ship, with little stability reserve. The IMO empirical form was adopted rather than the pendulum expression T = 2π × radius of gyration ÷ square root of (g × GM), because the roll radius of gyration is rarely known on board — which is exactly what the coefficient C estimates from hull geometry. Enter the beam, the draught, the waterline length and the metacentric height GM.

Moulded beam B (m) Mean draught d (m) Waterline length L (m) Metacentric height GM (m)

Result

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Ship Roll Period by the IMO Formula: Stiff or Tender?

Roll period is the quickest way to feel how stable a ship is without opening the stability booklet. Master and chief officer reach for it when signing off a cargo plan, the naval architect uses it inside the IMO weather criterion, and on board it becomes a test in itself: time the roll and back out the real metacentric height of the loaded condition. Too much GM gives a short period and violent roll acceleration that breaks lashings and hurts crew; too little leaves the ship sluggish and slow to right herself.

The IS Code 2008 formula reads T = 2 × C × B ÷ √GM, where C = 0.373 + 0.023 × (B ÷ d) − 0.043 × (L ÷ 100). B is the moulded beam, d the mean draught, L the waterline length and GM the metacentric height, all in metres, with T in seconds. On the sample values — 18 m beam, 6.5 m draught, 95 m length and 1.2 m GM — the coefficient works out at 0.396 and the period at 13.01 s. We prefer this empirical form over the pendulum one, T = 2π × k ÷ √(g × GM), since the roll radius of gyration is rarely known on board, and estimating it from hull geometry is exactly what C does.

The GM going in has to be the free surface corrected value, not the solid one — slack tanks eat half a metre of GM and stretch the period with nothing else changed in the loading plan. L means waterline length, not overall length: feeding LOA inflates the coefficient a little and shortens the period. Bear in mind as well that C was fitted on conventional merchant hulls and that the expression describes small amplitude rolling in calm water with the vessel upright. Barges, unusually wide hulls, deep bilge keels and active anti-roll tanks all fall outside the calibrated range.

Frequently asked questions

How do I get GM from a timed roll period?

Invert the formula: GM = (2 × C × B ÷ T)². With the geometry of the example the product 2 × C × B equals 14.25, so a measured period of 15 s maps to a GM of 0.90 m, 13 s to about 1.20 m and 10 s to 2.03 m. This is the classic rolling period test: moor the ship slackly, pick a moment with no beam wind, time several complete oscillations and divide by the number of cycles. The answer stays approximate, yet it makes a solid cross-check against the GM the loading computer reports.

Is 13.01 s a good period for this ship?

It sits on the tender side for a 95 m hull. Useful anchors for this same geometry: an 8 s period corresponds to 3.17 m of GM, a stiff ship with high roll acceleration and poor comfort; 13 s matches the 1.20 m of the example, a common figure for a loaded vessel; and 20 s implies only 0.51 m, close to the minimum most rules demand. The test never rests on the period alone, but on whether the matching GM clears the stability curves required for that loading condition.

Does the formula suit every kind of vessel?

No. The C coefficient was derived from conventional merchant hulls, and the IMO itself treats the expression as an approximation. Very wide hulls relative to draught, barges and pontoons, multihulls, craft with deep bilge keels or an active anti-roll tank all sit outside the calibrated range, and the real period can drift a long way from the computed one. For those, a roll decay trial on the finished vessel or a direct calculation with a radius of gyration from the weight distribution gives a far better answer.

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Block Coefficient (Cb)

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Prismatic Coefficient (Cp)

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Constant-Rate Drying Period Time

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Gross Tonnage (GT — IMO 1969)

Calculates a vessel's gross tonnage GT with the formula from the 1969 International Convention on Tonnage Measurement of Ships: the moulded volume of all enclosed spaces multiplied by a coefficient that grows with the logarithm of that volume. GT is dimensionless, not a mass, and it drives port dues, crewing requirements and regulatory bracket. Enter the moulded volume.

The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.