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Metacentric height (GM)

Metacentric height, denoted GM, is the vertical distance between a ship’s centre of gravity G and its transmetacentre M. It is the primary index of initial transverse stability: a positive value means the vessel generates a righting moment when heeled to small angles, and a negative value means it will capsize without corrective action. GM governs the roll period, the response to cargo shifts, and compliance with the intact stability criteria in the IMO 2008 International Code on Intact Stability (Resolution MSC.267(85)). Every loaded or ballast condition entered in a ship’s stability booklet records a GM value corrected for free surface, and loading officers verify that value against the minimum permitted before departure. The ShipCalculators.com-hosted metacentric height calculator and companion tools cover the full chain from raw dimensions to verified stability margins, supporting deck officers, naval architects, and port state control surveyors working across all vessel types.

Contents

Background and history

The geometric description of a floating body’s stability was first placed on a rigorous footing by the French mathematician Pierre Bouguer in his 1746 treatise Traité du navire. Bouguer introduced the concept of the metacentre as the point about which a slightly heeled vessel appears to rotate, and he showed that stability depends on the vertical separation between that point and the centre of gravity. Within the same decade Leonhard Euler refined the hydrostatic theory in Scientia Navalis (1749), deriving the relationship between the metacentric radius and the second moment of the waterplane area - a result that underpins the BM = IT / V formula still used today.

Practical application lagged theory for over a century. The catastrophic loss of HMS Captain in 1870, which capsized with 472 lives, exposed how inadequate the empirical rules of the time were for predicting the stability of iron warships. The subsequent committee of inquiry by the British Admiralty established the practice of conducting inclining experiments on every new vessel and recording the resulting lightship KG in official documents. By the late nineteenth century the full hydrostatic calculation chain - from waterplane geometry to KM, and from inclining experiment to KG, leading to the computed GM - had become standard procedure in British and continental shipyards.

The twentieth century brought two reinforcing drivers: the formalisation of stability criteria and the development of computing methods for cross-curves of stability. The International Load Line Conference of 1930 linked freeboard directly to stability by requiring a minimum range of positive righting arm; later IMO work consolidated these requirements into Resolution A.749(18) (1993) and eventually the comprehensive IMO 2008 IS Code, adopted as Resolution MSC.267(85) by the Maritime Safety Committee on 4 December 2008 and entering into force on 1 July 2010. The 2008 code superseded A.749(18) for all ship types except those with specific sub-codes and remains the primary international instrument for intact stability.

Several catastrophic losses during the twentieth century reinforced the regulatory trajectory. The overturning of the ferry Herald of Free Enterprise in 1987 (193 deaths, bow doors left open), the capsizing of the Estonia in 1994 (852 deaths), and numerous smaller capsizings of fishing vessels and passenger ferries each prompted IMO review of the IS Code and SOLAS stability requirements. While the Herald and Estonia disasters involved damage stability (flooding) rather than inadequate initial GM, they prompted a wholesale review of stability assessment methodology and led to the enhanced probabilistic damage stability regime of SOLAS 2009. The broader regulatory context - including mandatory port state control inspections under the Paris MOU and Tokyo MOU - means that stability deficiencies, including loading conditions with GM below the approved minimum, are now among the detainable deficiencies checked by surveyors under the ISM Code and SOLAS convention frameworks.

Geometric foundations

Points and their vertical positions

Stability analysis uses a common vertical reference: the keel K. All heights are measured from K upward and are denoted by the prefix K.

The centre of buoyancy B is the centroid of the underwater displaced volume V. Its height KB rises as the vessel is loaded because the waterplane expands and the centroid shifts upward; for wall-sided hulls KB is approximately half the mean draught, though in practice values obtained from hydrostatic tables are more accurate.

The centre of gravity G is the centroid of all mass aboard - hull steel, machinery, cargo, ballast, liquids, and crew. Its height KG depends entirely on how and where mass is distributed, not on draught. Raising heavy cargo to upper decks increases KG; filling double-bottom tanks lowers it.

The metacentre M is a geometric construct. When a wall-sided hull is heeled to a small angle θ, the emerging and immersing wedges of buoyancy shift the centre of buoyancy from B to B’. The vertical line through B’ intersects the original vertical through B at the metacentre M. For angles small enough that the waterplane shape does not change materially (in practice, below about 10° to 15°), M remains essentially fixed in space, justifying the use of GM as a constant index.

The height KM = KB + BM, where BM is the metacentric radius.

The metacentric radius BM

The metacentric radius equals the second moment (moment of inertia) of the waterplane area about the vessel’s centreline IT divided by the displaced volume V:

BM = IT / V

IT carries units of m⁴ and V units of m³, giving BM in metres. Because IT increases as the beam cubed and V increases with draught, a wide shallow ship has a large BM, while a narrow deep ship has a small one. Container ships and bulk carriers exploit wide beams partly for this reason.

The waterplane second moment can be computed numerically using Simpson’s rule applied to the waterplane. The BM geometry calculator implements the IT / V relationship directly given waterplane half-breadths and displacement. Formula details appear on the BM from waterplane inertia formula page.

For longitudinal stability the analogous quantity is BML = IL / V, where IL is the second moment of the waterplane about a transverse axis through the centre of flotation. BML is typically 50 to 100 times larger than BM, which is why ships are much stiffer in pitch than in roll.

GM and the righting arm GZ

The metacentric height is:

GM = KM − KG

A positive GM means M lies above G, which is the stable case. A negative GM means G has risen above M; the vessel will loll to one side, and if uncorrected will capsize.

For small angles of heel θ (below about 10°), the righting lever is:

GZ ≈ GM × sin θ

At θ = 10° this gives GZ ≈ 0.174 × GM. The approximation deteriorates rapidly beyond 15° as the hull geometry changes and the wall-sided assumption no longer holds.

For larger angles, GZ is determined from cross-curves of stability (KN curves) stored in the stability booklet:

GZ = KN − KG × sin θ

where KN is the righting lever measured from the keel to the perpendicular from N (the intersection of the line of buoyancy with the centreline plane), tabulated at various displacements and angles. The GZ from KN cross-curve calculator automates this step given the ship’s KG and displacement. Detailed formula derivation is at the GZ from KN formula page.

Negative GM and loll

A vessel with negative GM at small angles may nonetheless exhibit positive righting arm GZ at some larger angle, provided the hull form provides a sufficiently large restoring moment through the shift of the centre of buoyancy. This is the condition of loll: the ship lies stably at an angle of heel on one side (or alternately on either side, shifting with each roll) where the wall-sided geometry has restored GZ to zero. The loll angle θL for a wall-sided vessel can be estimated from:

tan²θL = −2 × GM / BM

(noting that GM is negative, making the product positive). Loll is not a stable equilibrium in the usual sense - any additional heeling beyond the loll angle may cause rapid capsize because GZ decreases. The corrective action for loll is always to lower the centre of gravity by adding ballast low in the ship, never to add mass high in the ship or to open cross-flooding valves without careful consideration. The wallsided heel calculator allows estimation of the loll angle and the GZ curve shape in the wall-sided regime.

The metacentric height in the context of trim

The metacentric height applies to a vessel floating at its equilibrium waterline. If the vessel is trimmed - that is, floating at different draughts at bow and stern - the metacentric height is technically the value pertaining to the trimmed waterplane and trimmed displaced volume. In practice, hydrostatic tables are constructed at even keel and corrections are applied for trim. For large trims (exceeding approximately 0.5% of LBP) the corrections can become significant, particularly for KB and BM, and naval architects sometimes compute a separate set of trimmed hydrostatics for typical operational conditions. The mean draught calculator and the density and draught calculator assist in establishing the correct displacement at the actual waterline.

Transverse and longitudinal metacentric height

Transverse GM

The transverse metacentric height GMT is almost always what is meant by the unqualified term GM. It governs rolling behaviour and is subject to the IMO minimum criteria described in a later section.

Longitudinal GM and MCT1cm

The longitudinal metacentric height GML is vastly larger than GMT for any practical hull - typically 100 to 300 times the ship’s length for full-form vessels. Its practical use is in computing the moment to change trim 1 cm (MCT1cm):

MCT1cm = (Δ × GML) / (100 × LBP)

where Δ is displacement in tonnes and LBP is length between perpendiculars in metres. MCT1cm has units of tonne-metres per centimetre. It tells the officer how much trimming moment must be applied - by shifting or adding weights - to alter the ship’s trim by one centimetre. The MCT1cm calculator and its formula page cover the derivation and worked examples.

A common operational application: if a weight of mass m at distance d from the centre of flotation is added, the change in trim is (m × d) / MCT1cm centimetres. The resulting draught changes at bow and stern depend on the distances from the centre of flotation to each perpendicular. The trim from weight shift calculator and the trim from loading centroid calculator handle these computations.

The inclining experiment

The inclining experiment is the only direct method of determining a ship’s actual KG and hence GM. It is mandatory for all new builds before entering service and must be repeated if major structural modifications alter the lightship weight or its distribution by more than specified tolerances.

Procedure

The experiment is conducted afloat with the ship in as light and complete a state as possible, with all tanks sounded and all temporary items recorded. A known mass m is shifted transversely through a measured distance d, causing the ship to heel. The heel angle θ is read from a pendulum (plumb bob) of known length Lp with deflection x, giving tan θ = x / Lp. From static equilibrium:

GM = (m × d) / (Δ × tan θ)

where Δ is the total displacement at the time of the experiment. Multiple transverse shifts in both directions are performed and the results averaged. The inclining experiment calculator and the inclining experiment pendulum check calculator implement this formula. Worked derivation is at the GM from inclining experiment formula page.

IMO IS Code requirements for inclining experiments

Chapter 7 of the IMO 2008 IS Code (Resolution MSC.267(85)) sets out the conditions under which an inclining experiment must be carried out and accepted. Key provisions include:

  • The experiment must be witnessed or approved by the flag state administration or a recognised organisation on its behalf.
  • Weather conditions must allow the ship to float freely without excessive motion; wind speed should not exceed approximately force 3 Beaufort.
  • All significant items of variable mass (fuel, water, provisions) must be as small as possible or fully accounted for, with tanks either pressed full or pressed empty where practicable.
  • A minimum of four weight movements are required, and the resulting GM values must be consistent within specified tolerances before a mean is accepted.
  • The lightship displacement and KG derived from the experiment form the basis of all subsequent stability calculations for the life of the ship.

The stability booklet required under SOLAS Chapter II-1 Regulation 5-1 must be based on the inclining experiment data, updated after major modifications.

Derivation of KG from displacement survey

If an inclining experiment is not practicable (for example, on a sister ship), a displacement survey combined with an analytical estimate of KG from individual component weights may be accepted by some administrations. However, the inclining experiment remains the preferred and legally more defensible basis.

Periodic re-inclining and lightship surveys

The IMO 2008 IS Code and most flag state regulations require a lightship check (either a full re-inclining experiment or a deadweight survey) at intervals not exceeding five years for passenger ships, and at major drydockings for cargo ships. The purpose is to detect the gradual accumulation of mass from modifications, additions, removed equipment, and the weight gain from accumulated paint layers, sea growth removed at drydock, and stores inventory discrepancies. A ship whose actual lightship weight drifts significantly from the recorded value will have a KG that departs from the stability booklet assumptions, potentially placing all loaded conditions outside their approved limits without any indication on the loading computer. Classification societies increasingly require periodic stability audits as part of their survey cycle.

Uncertainty in the inclining experiment

The principal sources of uncertainty in an inclining experiment are: inaccuracy in the measured displacement (arising from errors in draught readings and the density of the water), uncertainty in the waterplane area used to correct for trim, incomplete accounting for all items on board, pendulum-reading error, and effects of wind, current, or mooring lines restraining the ship during the heeling measurements. The IS Code requires that results from different shifts be consistent within a specified tolerance; if they are not, the experiment must be repeated or the outliers investigated. A well-conducted experiment achieves a KG uncertainty of approximately ±0.02 m to ±0.05 m for a large cargo ship, but errors of ±0.10 m or more are possible in adverse conditions.

Free surface effect and the free surface correction

Liquid in a partially filled tank shifts transversely when the ship heels, moving the centre of gravity of that liquid outward and thereby reducing the effective righting moment. This is the free surface effect.

The free surface correction (FSC) to GM is:

FSC = (ρL × iT) / (Δ)

where ρL is the density of the liquid in the tank (t/m³), iT is the second moment of the free surface area of the tank about its own centreline (m⁴), and Δ is the ship’s displacement (tonnes). The corrected effective metacentric height is:

GMeff = GMsolid − FSC

Multiple tanks each contribute their own FSC term, and the total correction is the sum of all individual terms regardless of whether the tanks are interconnected or not (the moment of inertia is computed per compartment for intact cases).

Practical implications are significant. A double-bottom fuel oil tank 20 m long and 10 m wide with density 0.95 t/m³, when half-full on a 10,000-tonne vessel, contributes approximately 0.16 m reduction in effective GM. Ballast tanks in wings at deck level can reduce effective GM by 0.3 m or more on larger vessels. The free surface correction calculator quantifies this for a single tank and the formula page for free surface correction presents the derivation.

Operational countermeasures include pressing tanks as full as possible, dividing tanks with longitudinal divisions (which reduce iT by a factor of 8 for two equal halves), and sequencing ballast exchanges to avoid large free surfaces simultaneously. The free-surface effect wiki article examines these in depth.

IMO 2008 IS Code minimum criteria

Intact stability criteria (IS Code 2.2)

The IMO 2008 IS Code Chapter 2.2 specifies minimum requirements for the GZ curve (righting arm curve) that every loading condition must satisfy:

  • The area under the GZ curve from 0° to 30° shall not be less than 0.055 m·rad.
  • The area under the GZ curve from 0° to 40°, or from 0° to the angle of downflooding if that angle is less than 40°, shall not be less than 0.090 m·rad.
  • The area under the GZ curve between 30° and 40°, or between 30° and the downflooding angle, shall not be less than 0.030 m·rad.
  • The righting arm GZ shall be at least 0.200 m at an angle of heel of 30° or greater.
  • The maximum righting arm shall occur at an angle of heel not less than 25°, and preferably not less than 30°.
  • The initial metacentric height GM0 shall not be less than 0.150 m.

These criteria apply to all ships of 24 m in length and above unless a more specific instrument applies (for example, the High Speed Craft Code or the MODU Code). They represent minimum thresholds, not design targets; a ship that just satisfies all six criteria may still have inadequate stability in severe sea conditions. The dynamic stability area calculator computes the areas under the GZ curve numerically. The intact stability KG limit calculator converts a required minimum GM or GZ into a maximum allowable KG at each draught.

Weather criterion (IS Code 2.3)

The weather criterion tests the vessel’s ability to withstand the combined effect of a beam wind and rolling in waves. In the Severe Wind and Rolling Criterion:

  1. A steady heeling moment from a wind pressure of 504 Pa acting on the lateral windage area is applied.
  2. The vessel is assumed to roll from the resulting steady heel angle windward by the amplitude of parametric or forced rolling.
  3. The area of the righting arm curve from the windward roll amplitude to the downflooding angle or 50° (whichever is less) must exceed the area of the heeling arm curve from that same angle to the steady wind heel.

This criterion effectively sets a lower bound on the dynamic stability reserve and on the range of positive GZ. For vessels with large windage areas - passenger ships, car carriers, ro-ro ferries - the weather criterion is often the governing constraint rather than the area criteria above.

The rolling amplitude assumed in the weather criterion is calculated from empirical coefficients that account for bilge keel area, breadth-to-draught ratio, and block coefficient. The formula uses tabulated factors (X1, X2, k, and r) derived from systematic model tests. A vessel with large bilge keels, high block coefficient, and moderate beam-to-draught ratio will have a lower assumed rolling amplitude, which relaxes the weather criterion. A fine-hulled vessel with small bilge keels and high beam-to-draught ratio may require substantially more dynamic stability reserve to pass the criterion.

Second generation intact stability criteria

The IMO Maritime Safety Committee has been developing a new generation of stability criteria (Second Generation Intact Stability Criteria, SGISC) since the early 2000s. These address five stability failure modes not covered by the 2008 IS Code: pure loss of stability in waves (reduction of GM in wave crests), parametric rolling, dead ship condition (loss of propulsion in beam seas), surf-riding and broaching, and excessive acceleration. The criteria use both Level 1 and Level 2 vulnerability checks plus direct assessment. MSC.1/Circ.1627 and subsequent circulars have progressively refined these criteria. As of 2025, adoption in SOLAS is anticipated for newbuildings, with a particular focus on container ships and large bulk carriers exposed to parametric rolling risk. The parametric roll susceptibility calculator implements the Level 1 vulnerability check for parametric rolling.

Special criteria for specific vessel types

The IMO 2008 IS Code contains additional criteria for passenger ships (resolution of the vessel after passenger crowding), cargo ships carrying grain (IS Code Annex, see also the grain heel calculator), offshore supply vessels, mobile offshore drilling units, pontoons, and vessels in the De-icing Code. Flag state administrations may impose stricter requirements. Classification societies such as Lloyd’s Register, DNV, Bureau Veritas, and the American Bureau of Shipping publish their own loading manual requirements that often require explicit verification of all IS Code criteria for each loading condition.

Onboard loading computers and stability software

Modern cargo ships and all passenger ships above specified sizes are required by SOLAS Chapter II-1 to carry an approved onboard loading instrument or computer. The instrument calculates loading condition stability in near-real time, taking the current weights and positions of cargo, ballast, fuel, and stores and computing displacement, KG, effective GM (after free surface corrections), and the GZ curve, then verifying all IS Code criteria.

Classification society approval of the loading computer involves checking the computed values for a set of benchmark loading conditions against the approved stability booklet values; the tolerance is typically within ±1% or ±0.01 m for GM and within ±2% for displacement. Surveyors from port state control authorities check that the loading computer certificate is current, that the software version matches the approved version, and that the current loading condition as shown on the instrument complies with the approved stability booklet.

Loading computers have evolved from dedicated hardware units, which were common from the 1980s onwards, to software running on standard PC platforms and, since the early 2010s, to web-accessible cloud-based loading platforms used by larger fleet operators. The underlying computation remains the same: the ship’s lightship KG and displacement from the inclining experiment, combined with a database of cargo and ballast positions, fluid density corrections, and free surface moment tables, are used to compute the current KG, from which effective GM is derived using hydrostatic tables interpolated for the current mean draught. The metacentric height calculator replicates the core GM = KM − KG step for manual verification.

Typical GM ranges by ship type

The appropriate magnitude of GM varies by vessel type, mission, and regulatory category. Values that are too low risk capsize; values that are too high produce a short, snapping roll that places excessive stress on cargo, lashings, and crew.

Passenger ships typically operate with GM between 0.15 m and 0.30 m. The stability criteria for passenger ships include requirements related to passenger crowding on one side, and the weather criterion is stringent given large superstructures. Very low values are therefore avoided, but a high GM makes the rolling motion uncomfortable and dangerous for passengers.

Ro-ro vessels and car carriers target GM in the range of 0.5 m to 1.0 m. The relatively low values reflect the high centre of gravity imposed by multi-deck vehicle cargo and the need to avoid a violent roll that could shift or damage vehicles. The ro-ro vessel wiki article discusses stability considerations in depth.

Bulk carriers commonly operate with GM between 1.0 m and 3.0 m. Homogeneous cargoes loaded in holds produce a low centre of gravity when holds are full, but heavy grain cargoes can shift and reduce effective GM; see the grain heel calculator and the IMSBC Code article for cargo securing requirements. The bulk carrier article covers operational loading patterns. Large GM values cause rapid rolling that can fatigue structural details at hatch corners.

Container ships typically have GM in the range of 1.0 m to 2.5 m. The wide beam of modern ultra-large container vessels (ULCV) tends to produce a large BM and thus a high KM, but stacking heavy containers on deck raises KG considerably. Parametric rolling, a phenomenon driven partly by variation in waterplane GM in head seas, is a specific hazard for fine-hulled container ships; the parametric roll susceptibility calculator assesses this risk.

Oil tankers have GM values typically in the range of 1.0 m to 4.0 m, depending on draught and the extent of ballast. Full-load conditions tend toward moderate GM because cargo fills the double bottom as well as cargo tanks, concentrating mass low. Ballast conditions are more variable; a tanker in ballast with partially filled ballast tanks may have large free surfaces unless tanks are pressed. The oil tanker article and the free-surface effect article address these conditions.

Chemical tankers face additional complications because cargo segregation requirements may leave several partially filled tanks simultaneously, each contributing free surface corrections. See the chemical tanker article and the IBC Code article.

LNG carriers present a distinctive case because LNG cargo tanks are either pressed full or empty in normal service (partial filling is restricted to sloshing-certified tank types), which limits free surface effects from cargo. Heel from LNG heel-return ballasting is managed carefully; the LNG heel return calculator quantifies the associated list.

Fishing vessels are among the most statistically vulnerable vessel types for stability failures. Many operate with GM values that are adequate in calm water but become marginal after catching a large quantity of fish, taking on deck water, deploying nets that create a large windage area, or when large quantities of ice accumulate on rigging. The IMO Torremolinos International Convention for the Safety of Fishing Vessels (1993) and its successor the 2012 Cape Town Agreement set out stability requirements for vessels 24 m and above, but most fishing vessel losses still involve craft below this length that operate without formal stability assessment.

High speed craft (HSC), including fast ferries and patrol vessels, typically have beam-to-draught ratios larger than conventional vessels and may rely on dynamic lift (planing, hydrofoil, or SWATH configurations) for a component of their support. The IMO HSC Code 2000 specifies alternative stability criteria based on the dynamic heel angle and the relationship between static and dynamic GM. At speed, the effective GM of a planing craft differs from its hydrostatic value because a proportion of the weight is supported by hydrodynamic lift rather than buoyancy.

The figures quoted above for each vessel type are typical values from loaded departure or full-ballast conditions and are subject to wide variation. Design values must be established through the approval process; operational values are verified for each specific loading condition using the stability booklet and loading computer before departure.

Roll period and GM

Natural roll period

The natural period of transverse rolling Troll is related to GM by:

Troll = 2π × k / √(g × GM)

where k is the radius of gyration of the vessel about the longitudinal axis through G (m) and g is gravitational acceleration (9.81 m/s²). Because the radius of gyration is not directly measurable without an inclining experiment performed in the rolling mode, simplified empirical formulas are preferred on board.

Weiss and similar empirical formulas

The most commonly used approximation in practice is the simplified Weiss formula:

Troll ≈ 0.78 × B / √(GM)

where B is the beam of the vessel in metres and Troll is in seconds. The constant 0.78 is an empirical average for steel ships; some authorities use 0.80 for vessels with significant superstructure or 0.73 for low-superstructure tankers and bulk carriers. The formula is not accurate to better than roughly ±10% but is adequate for on-board checks and for estimating GM from an observed roll period - the reverse calculation:

GM ≈ (0.78 × B)² / Troll²

This reverse application is operationally useful: if rolling is observed to be unusually slow (long period), GM may be lower than expected. The GM from rolling period calculator implements both directions of this relationship. The natural roll period rational formula calculator uses the full gyration-radius approach.

Stiff and tender ships

A stiff ship has a high GM and consequently a short roll period. The rolling motion is rapid and the acceleration forces are high, which is uncomfortable for crew, potentially damaging for cargo, and structurally fatiguing. Bulk carriers in ballast condition and tankers at light draught sometimes exhibit stiff behaviour with GM values above 3 m.

A tender ship has a low GM and a long, slow roll. While more comfortable, a tender ship recovers slowly from heel and has less margin above the minimum criteria. Officers should monitor conditions that reduce GM - progressive flooding, free surface growth, consumption of ballast without topping up, or loading of top-heavy deck cargo. The minimum GM0 of 0.15 m in the IS Code represents the outer boundary of acceptable tenderness.

A ship with GM very close to zero will exhibit a persistent list or loll - lying at an angle where the large-angle geometry provides a righting moment even though the initial metacentric height is negative. A lolling ship must not be corrected by filling high tanks (which would add free surface and make the loll worse); instead, ballast should be added to the double bottom or cargo redistributed to lower the centre of gravity.

Roll period as a sea check

In the absence of an onboard loading computer or when a cross-check of calculated values is desired, an officer can estimate GM by timing the natural roll period. The ship is allowed to roll freely in calm to moderate sea conditions, and the period is measured over multiple complete oscillations and averaged. Using the simplified Weiss formula GM ≈ (0.78 × B)² / Troll², the implied GM can be compared with the stability booklet value for the approximate displacement. A significant discrepancy - for example, an observed roll period 30% longer than expected, implying GM roughly half the booklet value - should prompt an immediate investigation into the loading condition, free surface corrections, and any changes made since departure. This technique is limited in accuracy and unsuitable in heavy rolling conditions, but it has identified stability problems on vessels where the loading computer has been bypassed or incorrectly operated.

The roll resonance in following or quartering seas is a separate phenomenon: when the wave encounter period equals the natural roll period, resonant rolling can develop even in moderate wave heights. Reducing speed or altering course to change the encounter frequency is the standard operational response. The relationship between roll period and wave encounter frequency is fundamental to the parametric roll phenomenon, where the roll period is approximately twice the wave encounter period and the GM variation as waves pass under the ship drives an oscillating instability. See the parametric roll susceptibility calculator for details.

Loading conditions and stability booklet

SOLAS Chapter II-1 Regulation 5-1 requires every cargo ship of 24 m in length and above to carry an approved stability booklet that presents hydrostatic data, cross-curves of stability, and verified stability calculations for a representative range of loading conditions.

Standard conditions that must be included are:

  • Lightship condition: the ship with no cargo, stores, or ballast, as derived from the inclining experiment.
  • Departure condition, full load: at summer load line draught with full cargo, full bunkers, full stores, and full fresh water.
  • Arrival condition, full load: cargo as departure but bunkers and stores at ten per cent remaining.
  • Departure condition, ballast: no cargo, with sufficient ballast to achieve adequate trim and GM, bunkers full.
  • Arrival condition, ballast: as ballast departure but bunkers at ten per cent.
  • Additional partial cargo or homogeneous loading conditions as appropriate to typical service.

Each condition records displacement, KG solid (before free surface correction), total free surface correction, effective GM, and verification of all six IS Code criteria. The stability booklet loading condition template allows verification of limiting KG at any draught.

Changes to cargo, ballast, fuel, or stores between these tabulated conditions require the officer to perform an intermediate stability calculation, typically using the ship’s onboard loading computer approved under SOLAS. The new KG from weight addition calculator, KG from weight removal calculator, and KG from weight shift calculator support manual or cross-check calculations.

Operational management of GM

Ballast and cargo distribution

The most direct lever available to an officer for adjusting GM is the choice of which ballast tanks to fill and to what level. Filling double-bottom tanks lowers G and raises B (by increasing draught), both of which increase GM. Filling wing tanks high in the ship raises G and has a larger free surface effect per tonne, reducing effective GM. Pressing tanks fully eliminates free surface at the cost of flexibility.

Cargo distribution on container ships is managed through stowage planning software that enforces KG limits and stack weight restrictions simultaneously. The metacentric shift calculator quantifies the shift in G resulting from moving a weight from one position to another.

Interaction with trim

Changes in ballast that affect GM also affect trim. The longitudinal centre of gravity shift determines trim change via MCT1cm, while the vertical shift determines GM change. Optimising both simultaneously requires iterative calculation. The trim moment calculator and the MCT1cm calculator are used together for this purpose.

Trim itself affects KM by altering the waterplane geometry: for most hulls, trimming by the stern reduces KM slightly because the stern waterplane is finer and the bow waterplane is fuller, shifting the waterplane centroid aft and reducing the second moment about the centreline. The effect is typically small (of the order of a few centimetres) but should be checked in the hydrostatic tables when large trims are contemplated.

Icing allowance

In cold-weather operations, ice accumulation on exposed decks, rigging, and superstructure raises the centre of gravity and reduces GM. The IS Code Annex Chapter 5 (Icing Code) prescribes added mass allowances per square metre of projected area. The icing allowance calculator applies these corrections to a given loading condition. Ships operating in polar regions are additionally subject to the Polar Code; see the Polar Code article.

Grain cargo

Grain cargo presents a specific GM risk because the free surface effect of a grain surface behaves differently from a liquid surface - the grain can shift as a solid mass if it is not trimmed level and restrained. The SOLAS Chapter VI and IMSBC Code requirements mandate that grain carriers demonstrate a minimum residual GM of 0.300 m throughout the voyage after an assumed void at the top of each filled compartment. The assumed shift heeling moment is computed from the grain loading geometry using the volumetric heeling moment tabulated in the Grain Loading Document. The grain heel calculator implements the assumed shift moment approach. See also the IMSBC Code article and the bulk carrier article.

Heavy weather operational guidance

In severe weather, operational decisions about ballasting, speed, and course relate directly to GM and roll behaviour. A vessel rolling heavily in beam seas may benefit from increasing GM (by transferring ballast to double-bottom tanks) to shorten the roll period and reduce rolling amplitude, particularly if the current period is close to the dominant wave period. Conversely, a stiff ship that is slamming heavily in head seas may benefit from a small reduction in GM achieved by topping up a high wing tank, lengthening the roll period and reducing structural acceleration loads.

Cargo shift is a significant hazard when GM is high and accelerations are large. Container stacks, bulk cargoes with low angle of repose, and vehicles on ro-ro decks are all susceptible to shifting when roll accelerations exceed a threshold. The cargo securing manual required under the CSS Code specifies lashing arrangements and maximum allowable accelerations that are predicated on the vessel’s roll period and hence on GM. A vessel that departs with lower GM than recorded in the stability booklet - and therefore a longer roll period than assumed - may subject cargo lashings to lower accelerations in beam seas but may also provide less restoring moment during a large heel event.

Reserve buoyancy and downflooding angle

The downflooding angle - the angle of heel at which sea water can enter the vessel through openings in the hull or superstructure - directly limits the range of stability used in the IS Code area criteria. A vessel with high freeboard and a high downflooding angle benefits from a larger range of positive GZ, while a vessel with low freeboard, open freeing ports, or low-sited ventilators has its stability range truncated. This is one mechanism by which the load line requirements interact with stability: reducing freeboard lowers the downflooding angle and thus reduces the area under the GZ curve that can be counted toward the IS Code criteria. The reserve buoyancy calculator quantifies the available freeboard margin above the waterline for a given loading condition.

Passenger crowding

Passenger ships must show that heeling from all passengers crowding to one side does not reduce GM below zero and does not produce excessive list. The passenger crowding heeling moment calculator quantifies the heeling arm for a given passenger distribution. The SOLAS convention article summarises the underlying regulatory framework.

GM in loading condition documentation

Stability information required by port state control

Port state control officers (PSCOs) under the Paris MOU, Tokyo MOU, and other regional agreements have authority to inspect a vessel’s stability documentation and loading condition as part of an expanded inspection. A PSCO may check that the stability booklet is approved and on board, that the loading computer has a valid approval certificate, and that the current loading condition as displayed on the loading computer or calculated manually shows GM (corrected for free surfaces) at or above the approved minimum. If the actual condition cannot be verified - for example, because the loading computer is inoperative and no manual calculation has been performed - the vessel may be detained until the stability is confirmed.

The ISM Code requires the ship’s Safety Management System to include procedures for verifying stability before departure. A vessel that departs without performing a stability check for the current loading condition is in violation of its SMS, regardless of whether the actual GM is adequate. Classification societies verify SMS compliance during safety management audits.

Stability data maintenance and the stability booklet

The stability booklet is a controlled document: any change to the approved lightship data, the loading conditions, or the hydrostatic tables requires classification society re-approval. Even minor physical modifications to the ship - installation of new equipment on deck, addition of a deck crane, removal of a mast - require a lightship assessment to determine whether the change in KG and displacement is within the tolerance permitted without a full re-inclining.

Shipyards maintain a stability file for each vessel built, which includes the original design hydrostatics, the inclining experiment report, and the approved stability booklet. The file is transferred to the ship owner and must be updated and retained throughout the ship’s operational life, including for sale to subsequent owners. At demolition, stability records form part of the inventory of hazardous materials required by the Hong Kong Convention.

Electronic stability booklets and class approval

Since approximately 2010, many classification societies accept electronic stability booklets - PDF versions of the approved document stored on the ship’s network and accessible from the loading computer. The IMO Maritime Safety Committee has issued guidance (MSC.1/Circ.1380, 2010) on the use of electronic stability instruments, including requirements for data integrity, backup, and update procedures. Electronic booklets are considered equivalent to paper books provided the document management system ensures that only the current approved version is accessible and that any superseded version is clearly marked.

Relationship to large-angle stability and damage stability

GZ curve and dynamic stability

The static GZ curve is the complete record of a vessel’s righting arm at all angles from 0° to capsize. Key features are:

  • The initial slope, which equals GM for small angles (the tangent at the origin has gradient GM in m per radian).
  • The maximum GZ value and the angle at which it occurs.
  • The range of stability - the angle at which GZ returns to zero after passing the maximum.
  • The area under the curve, representing the energy absorbed before capsize.

The angle of vanishing stability (AVS) is the upper limit of the range. Ships with low GM but wide beam may have a short initial range but develop positive GZ again at large angles owing to deck edge immersion - a feature captured by the wall-sided stability formula. The wallsided heel calculator extends the small-angle approximation into the region where the wall-sided assumption holds but the metacentric height approximation does not.

Dynamic stability, the area under the GZ curve, determines how much energy from wave action can be absorbed without capsizing. A ship can survive a wave that heels it past the maximum GZ if the kinetic energy of the roll does not exceed the dynamic stability reserve between the maximum GZ angle and the AVS. This is the physical basis of the weather criterion’s area comparison test. The dynamic stability area calculator computes the relevant areas using numerical integration.

For a more detailed treatment of large-angle behaviour, see the intact stability article.

Connection to damage stability

When a compartment is flooded, the vessel’s displacement, draught, and trim change instantaneously (lost buoyancy method) or the waterplane area shrinks (added weight method). Either way, both KB and BM change, and the ingress of water also changes KG if the flooded space is above the original keel. The net result is a residual GM that must meet minimum criteria under SOLAS probabilistic damage stability rules (Regulation II-1/7 for passenger ships and II-1/6 for cargo ships). The residual GM checker verifies whether the post-damage condition satisfies these requirements. A comprehensive treatment appears in the damage stability article.

Load line assignment is also intertwined with stability: a lower freeboard reduces the height of downflooding openings, lowering the effective range of stability and the area under the GZ curve. The load line article describes how the International Load Lines Convention 1966 and its 1988 Protocol set minimum freeboards that are partly predicated on the ship maintaining an adequate GZ range.

Effect of permeability on post-damage GM

The effective reduction in righting moment after flooding depends critically on the permeability of the flooded compartment - the proportion of the space that is actually occupied by water rather than by structural members, machinery, cargo, or other solids. A void tank has permeability close to 100%; a machinery space typically has permeability of about 85%; a cargo hold loaded with a solid bulk cargo may have permeability of 60% to 70%. Lower permeability means less water enters, which reduces the loss of buoyancy and the rise in KG, leaving a higher residual GM. The damage permeability and effective volume calculator is used to compute the effective flooded volume taking permeability into account.

Damage stability and the SOLAS probabilistic method

The probabilistic damage stability method introduced into SOLAS Regulation II-1 evaluates the overall survivability index A of the vessel as a summation over all credible two- and three-compartment flooding scenarios. Each scenario contributes a product of the probability p that the flooding extends to those compartments (a function of watertight subdivision geometry), the probability s that the vessel survives that flooding (a function of the residual GM and GZ range), and a factor accounting for the loading condition. The required overall index R depends on vessel type and length. The stability calculation for each damage case requires the same hydrostatic framework - KM, free surface corrections, and GZ curves - that governs intact stability.

Design considerations

Beam and depth ratios

Naval architects control the initial metacentric height primarily through the choice of beam, draught, and the vertical distribution of mass. The metacentric radius BM scales with the square of the beam-to-draught ratio; doubling the beam at constant draught quadruples BM and therefore dramatically increases KM. Conversely, a narrow, deep hull has low BM, and a high KM must then be achieved by keeping KG low - difficult if the cargo being carried is inherently top-heavy.

The waterplane coefficient Cwp quantifies how closely the waterplane approaches a rectangle. A full-form hull (high Cwp, as in tankers and bulk carriers) has a larger IT than a fine-form hull (lower Cwp, as in naval vessels and fast container ships) at the same beam and length. The waterplane coefficient calculator and the prismatic coefficient calculator help contextualise the form coefficients that underlie the hydrostatic calculation.

For most displacement vessels, KG is determined by structural weight, cargo type, and the arrangement of machinery and accommodation. Container ships face a particular tension: wide beam raises KM (and hence GM) but also raises cargo stacks higher, increasing KG. The net effect on GM depends on the actual KG increase per unit of beam increase, which is a function of stowage density and stack height. Ultra-large container vessels with beams exceeding 60 m achieve adequate GM despite very high KG values because KM rises steeply with beam.

Bilge keels and anti-rolling devices

Bilge keels are longitudinal fins welded to the hull at approximately the bilge turn. They do not affect GM or the static stability curve but increase roll damping, reducing the amplitude of resonant rolling. The effective roll period and hence the perceived tenderness of the ship is influenced by bilge keel area, but the static GM is unchanged. Anti-rolling tanks (passive or active), gyroscopic stabilisers, and fin stabilisers similarly affect roll amplitude without changing the hydrostatic GM. Their presence is relevant to the weather criterion rolling amplitude factor used in IS Code 2.3 calculations.

GM management during tank cleaning and ballast water exchange

Ballast water exchange and tank cleaning operations frequently require partial emptying of large tanks, creating free surface effects that temporarily reduce effective GM. The IMO Ballast Water Management Convention (in force since 8 September 2017) requires exchange or treatment of ballast water in designated ocean exchange zones. Sequential exchange (emptying one tank before refilling) minimises free surface at any moment but requires careful sequencing to avoid excessive trim or list. The ballast water management convention article covers the regulatory framework, and stability officers use the free surface correction calculator to monitor GM through each exchange step.

Hydrostatic context

GM does not exist in isolation from other hydrostatic quantities. The complete set of interrelated parameters typically tabulated against draught in a vessel’s hydrostatic tables includes:

  • Displacement Δ (tonnes), which drives the free surface correction denominator.
  • Tonnes per centimetre immersion (Tpc), from which changes in mean draught from added weight are calculated; see the TPC calculator and TPC from Simpson’s rule calculator.
  • MCT1cm, directly dependent on GML, as discussed above.
  • Centre of flotation (F), the pivot point about which trim changes occur.
  • KB, from which KM = KB + BM is derived.
  • BM, from waterplane inertia.
  • KM, the sum of KB and BM, tabulated against draught.
  • Free surface corrections for each tank, tabulated or calculated separately.

For a broader discussion of these tables and their application see the hydrostatics and Bonjean article. The trim and list article addresses the closely related topic of how weight movements change longitudinal and transverse attitude. Waterplane geometry itself - and the waterplane area coefficient - connects hull form to BM through the relationship IT = Cw² × LBP × B³ / 12 (an approximation for prismatic waterplanes). The block coefficient and hull form design articles contextualise the broader relationship between form and stability.

Calculator resources

The ShipCalculators.com calculator catalogue groups all stability tools under the intact stability section. Relevant calculators for metacentric height and related quantities include:

See also

References

  1. International Maritime Organization. International Code on Intact Stability, 2008 (2008 IS Code). IMO Resolution MSC.267(85), adopted 4 December 2008. London: IMO, 2009.
  2. International Maritime Organization. SOLAS Consolidated Edition. Chapter II-1, Part B, Regulations 5, 5-1, 6, 7. London: IMO, 2020.
  3. Bouguer, Pierre. Traité du navire, de sa construction et de ses mouvemens. Paris: Jombert, 1746.
  4. Euler, Leonhard. Scientia Navalis, Vol. I. St. Petersburg: Academia Scientiarum, 1749.
  5. Barras, C. B. Ship Stability for Masters and Mates, 7th ed. Oxford: Butterworth-Heinemann, 2012.
  6. Derrett, D. J., revised by Barrass, C. B. Ship Stability for Masters and Mates. Oxford: Elsevier, 2012.
  7. Molland, A. F. (ed.). The Maritime Engineering Reference Book. Oxford: Butterworth-Heinemann, 2008. Chapter 3, “Ship Hydrostatics and Stability”.
  8. International Maritime Organization. Guidelines for the Assessment of the Residual Stability of Passenger Ships after Flooding. MSC/Circ.1144, 2004.
  9. International Load Lines Convention 1966 and Protocol of 1988. IMO Publication IC110E. London: IMO.

Further reading

  • Rawson, K. J., and Tupper, E. C. Basic Ship Theory, 5th ed. 2 vols. Oxford: Butterworth-Heinemann, 2001. Chapters 5 and 6 cover metacentric height and large-angle stability from first principles with full derivations.
  • Lewis, E. V. (ed.). Principles of Naval Architecture, 2nd rev. ed. Vol. I, “Stability and Strength”. Jersey City, NJ: SNAME, 1988.
  • Kobylinski, L., and Kastner, S. Stability and Safety of Ships: Regulation and Operation. Amsterdam: Elsevier, 2003.