Ship Stability Explained: KG, GM, GZ Curves and Righting Moments

A ship may float perfectly upright and still have inadequate stability.

Floating only requires the vessel’s weight to equal the buoyancy force produced by the displaced water. Stability asks a different question: what happens after the vessel is disturbed?

Wind, waves, turning forces, cargo movement, lifting operations and uneven loading can all heel a vessel away from the upright position. Once that happens, the underwater shape changes, the centre of buoyancy moves and a moment develops. That moment may return the vessel upright, hold it at an inclined position or push it farther toward capsize.

Ship stability is therefore not represented by one number. It is better understood as a sequence:

1. KG and KM determine the initial metacentric height, GM.
2. GM determines the initial slope of the GZ curve.
3. The hull form determines how GZ develops at larger heel angles.
4. Freeboard, downflooding openings and watertight integrity determine how much of that stability can actually be used.
5. Loading, slack tanks and shifting weights continually change the result.

The International Maritime Organization’s 2008 Intact Stability Code brings together mandatory and recommended provisions covering GM, GZ, free surfaces, severe wind and rolling, watertight integrity and stability information for the master. The Code became mandatory under SOLAS and the Load Line Protocol in 2010.

This article explains the geometry behind the principal stability terms, shows why GM alone cannot describe overall stability and finishes with a complete calculation covering KM, corrected GM, GZ and righting moment.

What Is Transverse Ship Stability?

Transverse stability is a vessel’s ability to resist sideways inclination and develop a moment that tends to return it toward the upright position.

The term transverse refers to movement across the vessel, from port to starboard. It is different from longitudinal stability, which concerns trim by the bow or stern.

A transverse inclination can be caused by:

• Beam winds and waves
• Rudder and turning forces
• Uneven cargo or ballast distribution
• Cargo shifting
• Passengers moving to one side
• Suspended loads handled by shipboard cranes
• Towing or fishing gear pulling from an elevated or off-centre point
• Water collecting on deck
• Flooding or liquid movement inside the vessel

The vessel’s response depends on two principal forces:

• Weight, acting vertically downward through the centre of gravity, G
• Buoyancy, acting vertically upward through the centre of buoyancy, B

When the ship is upright and has no list, these forces normally act along the same vertical line. Their magnitudes are equal, so there is no rotational moment.

When the ship heels, the underwater volume changes shape. The centre of buoyancy moves toward the more deeply immersed side, separating the lines of action of weight and buoyancy. This separation creates a righting or capsizing moment.

The Main Stability Points: K, B, G, M and Z

Ship stability diagrams use several reference points. They do not all represent physical objects, and confusing their roles makes the subject much harder than it needs to be.

K — Keel Reference Point

K is the reference point from which vertical distances are measured.

It is normally placed at the moulded keel or another defined baseline used by the vessel’s hydrostatic data. K does not produce a force and does not move during an ordinary stability calculation.

The principal vertical measurements are:

• KB — height of the centre of buoyancy above keel
• KG — height of the centre of gravity above keel
• KM — height of the metacentre above keel

Using one consistent reference point allows these values to be combined directly.

G — Centre of Gravity

The centre of gravity, G, is the point through which the vessel’s total weight acts vertically downward.

Every item on board contributes to its position:

• Hull structure
• Machinery
• Cargo
• Ballast
• Fuel
• Fresh water
• Stores
• Crew and passengers
• Deck equipment
• Temporary loads

The vessel’s KG is calculated from vertical weight moments:

where:

• \(w\) is an individual weight
• \(kg\) is the vertical position of that weight above keel
• \(\sum w\) is the vessel’s total displacement

Adding a weight above the existing G raises KG. Adding a weight below G lowers it. Removing a low weight may raise KG, while removing a high weight may lower it.

A weight already on board moves G in the same direction as the weight itself. Shifting cargo to port shifts G to port. Raising a suspended load shifts the combined G upward toward the point of suspension.

Transport Canada’s stability training guidance describes G as the point through which the vessel’s mass acts and notes that its position changes with loading, fuel, ballast and cargo distribution.

B — Centre of Buoyancy

The centre of buoyancy, B, is the geometric centre of the vessel’s submerged volume.

The total buoyancy force acts vertically upward through this point.

Unlike G, which changes when weights are added, removed or moved, B changes mainly because the underwater geometry changes. It depends on:

• Draft
• Trim
• Heel angle
• Hull form
• Immersed appendages
• Watertight superstructure volume, where applicable

When an upright symmetrical vessel heels to starboard, more volume becomes immersed on the starboard side and volume emerges on the port side. The centre of buoyancy therefore shifts toward starboard, commonly from B to B₁.

That movement is what creates the separation between the weight and buoyancy forces.

A frequent misunderstanding is that B must always be above G for the vessel to be stable. This is not true. Many ships have G above B while upright. At small angles, the position of M relative to G, not B relative to G, determines initial stability.

M — Transverse Metacentre

At a small heel angle, the buoyancy force acts vertically upward through the shifted centre of buoyancy, B₁. If this new buoyancy vertical is extended upward, it intersects the original centreline at M, the transverse metacentre.

For small inclinations, M can be treated as approximately fixed. The distance between G and M is the metacentric height, GM.

The metacentre is a geometric construction rather than a permanent physical point. As heel becomes larger, the underwater shape changes substantially and the metacentric approximation becomes less reliable. The complete GZ curve must then be used instead of treating M as fixed.

Transport Canada’s engineering study guide defines M as the intersection between the buoyancy vertical and the vessel centreline at small inclination, and uses GM to describe the vessel’s initial stability.

Z — The End of the Righting Lever

Z is used to identify the horizontal lever between the vessel’s weight line through G and the buoyancy line through B₁.

The distance GZ is the righting arm or righting lever.

Z should not be treated as another centre inside the vessel. It is a point used to show the perpendicular distance between two parallel force lines.

How KB, BM, KM and KG Produce GM

The transverse metacentric height is calculated from:

KM is obtained from:

Therefore:

These relationships separate stability into two broad influences:

• Hull and waterplane geometry, represented by KB and BM
• Weight distribution, represented by KG

This distinction is useful because a vessel may gain stability either by changing its geometry or by lowering its centre of gravity.

What Does KB Represent?

KB is the vertical distance from the keel reference, K, to the centre of buoyancy, B.

Its value depends on the shape of the immersed hull.

For a rectangular box-shaped section:

where \(T\) is the draft.

Real ship sections are not rectangular, so actual KB must be obtained from hydrostatic data. Rules such as \(KB = 0.5T\), \(0.53T\) or similar expressions are only estimates for preliminary work and classroom calculations.

As draft changes, both the submerged volume and its geometric centre change. KB must therefore correspond to the vessel’s actual displacement, draft and trim.

What Does BM Represent?

BM is the transverse metacentric radius: the vertical distance between B and M.

It is calculated from:

where:

• \(I_T\) is the second moment of area of the waterplane about the longitudinal centreline, in \(\mathrm{m^4}\)
• \(\nabla\) is the displaced volume, in \(\mathrm{m^3}\)

If displacement mass and water density are known:

where:

• \(\Delta\) is displacement in tonnes
• \(\rho\) is water density in \(\mathrm{t/m^3}\)

BM shows why beam has such a strong effect on initial transverse stability. For a rectangular waterplane:

Beam appears to the third power. Increasing waterplane breadth can therefore increase BM much more strongly than increasing waterplane length.

This is often called form stability. A wide vessel can obtain substantial initial stability from its hull geometry even without an exceptionally low centre of gravity.

By contrast, a vessel that relies on a low KG is often described as having greater weight stability.

The distinction is not absolute—real vessels use both—but it explains why a broad barge and a ballasted sailing vessel can both be stable for very different reasons.

Positive, Zero and Negative GM

GM determines the vessel’s initial response at or near the upright position.

Positive GM

When M is above G:

A small heel creates a positive righting arm. The resulting moment tends to return the vessel upright.

This is stable equilibrium at the upright position.

Positive GM does not automatically prove compliance with intact stability requirements. It only establishes that the initial slope of the stability curve is positive.

Zero GM

When M and G coincide:

At very small angles, no initial righting arm is produced.

The vessel is in neutral equilibrium under the metacentric approximation. If it is inclined slightly, it has no immediate tendency either to return upright or to heel farther.

Hull geometry at larger angles may eventually create positive or negative GZ, but the upright condition itself has no initial restoring stiffness.

Negative GM

When G is above M:

A small inclination produces a moment that acts in the direction of heel rather than toward upright. The upright condition is unstable.

A vessel with negative initial GM does not always capsize immediately. Depending on its hull geometry, B may move sufficiently at a larger heel angle to produce a new equilibrium. The vessel can then settle at an angle of loll.

This condition is dangerous because the vessel may roll from one side to the other, has little useful stability reserve near upright and may respond unpredictably to free surfaces, wind or cargo movement.

Negative GM and an ordinary list are not the same condition.

GM Is the Beginning of the GZ Curve

For small heel angles:

where \(\phi\) is the heel angle.

For very small angles measured in radians:

so:

This reveals one of the most useful relationships in stability:

GM is the initial slope of the GZ curve.

A larger positive GM creates a steeper initial curve. A small positive GM creates a shallower initial curve. A negative GM makes the curve initially fall below zero.

However, GM does not define the complete curve. It does not show:

• Maximum GZ
• Angle of maximum GZ
• Range of positive stability
• Downflooding angle
• Area under the GZ curve
• Effect of deck-edge immersion
• Buoyancy from enclosed superstructures
• Stability behaviour at large heel angles

The equation \(GZ = GM\sin\phi\) is useful for initial stability, but it should not be extended across the entire heel range of a real ship.

Why the Metacentric Approximation Breaks Down

At a small heel, the change in underwater geometry is limited and M can be treated as nearly fixed.

At larger angles:

• The deck edge may enter the water
• Flared sections may become immersed
• Flat bottom areas may emerge
• Superstructure volume may begin contributing buoyancy
• Openings may approach the water
• The waterplane shape may change rapidly
• B follows a curved path rather than a simple lateral shift

M therefore does not remain fixed in the way assumed by the small-angle construction.

Finite-angle stability is normally evaluated using cross-curves or KN data. When KN is known:

If free surface correction is represented as a virtual rise of G, a corrected or virtual KG may be used:

and:

Approved loading computers normally perform these calculations using vessel-specific hydrostatic and stability data.

Reading a GZ Curve

A GZ curve plots righting arm against heel angle.

The horizontal axis represents heel angle in degrees. The vertical axis represents GZ in metres.

Several parts of the curve describe different aspects of the vessel’s stability.

Initial Slope

The slope near zero heel represents GM.

A steep initial rise indicates a relatively large GM. A shallow rise indicates a smaller positive GM.

Maximum GZ

The highest point on the curve is the maximum available righting lever for that loading condition.

The value of maximum GZ matters, but so does the angle at which it occurs. A curve that peaks very early may lose righting ability rapidly as heel increases.

Range of Positive Stability

The range of positive stability extends from the upright position to the angle at which GZ becomes zero again.

This second zero crossing is called the angle of vanishing stability.

Beyond this angle, GZ becomes negative and the vessel develops a capsizing moment.

The theoretical range must not be confused with a safe operating range. Downflooding may occur before the curve reaches zero.

Downflooding Angle

The downflooding angle is the heel angle at which an opening that cannot be closed weathertight becomes immersed.

Examples can include:

• Ventilator openings
• Open doors or hatches
• Air pipes
• Companionways
• Machinery-space openings
• Other unprotected accesses leading below deck

A calculated GZ curve may remain positive after downflooding begins, but that remaining part of the curve cannot be treated as dependable intact stability reserve. Water entering the hull changes displacement, free surface, KG and buoyancy geometry.

For this reason, stability criteria commonly use the downflooding angle where it is less than the specified limiting angle.

Area Under the GZ Curve

The area beneath the GZ curve is expressed in metre-radians.

It represents a measure of the vessel’s energy reserve against heeling. A narrow, high curve and a broad, moderate curve may have similar maximum GZ values but very different areas.

GM describes the curve close to upright. Maximum GZ describes its highest lever. The area describes how much restoring capacity is available across a range of heel angles.

This is why a single GM value cannot replace a complete intact stability assessment.

Righting Lever and Righting Moment

GZ is a distance. It does not by itself state how large the restoring moment will be.

The moment of statical stability is:

If displacement is entered in tonnes and GZ in metres, the result is commonly expressed as tonne-metres:

For a force-based result:

A 0.30 m righting lever produces very different moments on a 500-tonne vessel and a 50,000-tonne ship.

Transport Canada’s training material defines GZ as the separation between the weight and buoyancy force lines and gives the righting moment as displacement multiplied by GZ.

A positive GZ produces a righting moment. A negative GZ produces a capsizing moment.

Stiff and Tender Ships

The terms stiff and tender describe the vessel’s initial stability and rolling behaviour.

Stiff Vessel

A stiff vessel has a relatively large GM.

It develops a strong righting moment after a small heel and tends to have a shorter natural roll period. Its roll may feel quick and abrupt.

Possible consequences include:

• High transverse accelerations
• Uncomfortable motion for crew or passengers
• Larger forces on cargo lashings
• Increased racking loads
• Greater risk of cargo damage
• Violent motion in heavy beam seas

A large GM is therefore not automatically desirable.

Official UK Maritime and Coastguard Agency guidance warns that excessive initial stability can create rapid, violent motion and large forces on cargo and securing arrangements.

Tender Vessel

A tender vessel has a small but positive GM.

It tends to roll more slowly and may initially feel comfortable. However, it develops less righting moment at small angles and may heel farther under wind, turning forces or cargo movement.

A tender condition leaves less margin for:

• Additional free surface correction
• Fuel and ballast consumption
• Suspended loads
• Water on deck
• Cargo shifting
• Unplanned topweight
• Errors in KG or tank quantities

A slow roll does not prove that a vessel is safe. Likewise, a quick roll does not prove that it has adequate overall stability.

Roll behaviour also depends on hull form, mass distribution, radius of gyration, damping, bilge keels, loading condition and sea state.

The objective is not to maximise GM. It is to maintain a suitable GM and an acceptable complete GZ curve for the vessel’s design and operating condition.

Why High GM Can Coexist With Poor Overall Stability

Consider a broad, shallow vessel with a very large waterplane inertia.

Its BM may be large, creating a strong initial GM. The vessel will initially resist heel strongly.

However, if it has:

• Low freeboard
• Openings close to the waterline
• Limited reserve buoyancy
• A hull form that loses waterplane area rapidly
• A low downflooding angle

its GZ curve may peak early and terminate quickly.

The vessel can therefore be very stiff near upright but possess a limited range of usable stability.

The opposite can also occur. A vessel with moderate initial GM may develop a broad GZ curve and substantial stability reserve at larger angles.

The practical lesson is straightforward:

GM answers how strongly the vessel begins to return. The GZ curve answers how long that restoring ability continues.

Free Surface Effect and the Virtual Rise of G

A pressed-up tank contains no internal free surface. The liquid moves with the tank as part of the vessel.

In a partially filled tank, the liquid surface remains approximately horizontal as the vessel heels. Liquid moves toward the low side, creating a moment that reduces the ship’s righting ability.

The effect is commonly represented as a virtual rise in the vessel’s centre of gravity.

The free surface correction is:

where:

• \(\sum FSM\) is the total free surface moment in \(\mathrm{t\cdot m}\)
• \(\Delta\) is displacement in tonnes
• \(FSC\) is in metres

Corrected GM is:

For an ideal rectangular tank:

where:

• \(\rho_l\) is liquid density
• \(L\) is tank free-surface length
• \(B\) is tank free-surface breadth

Breadth is cubed. A wide slack tank can therefore produce a much greater free surface moment than a narrow tank of similar volume.

Transport Canada’s adequate stability and safety guidance defines free surface effect as a stability reduction caused by uncontrolled liquid movement and describes it as an equivalent or virtual rise in G. Current intact stability guidance requires free surface effects to be included in applicable loading conditions.

Three Free Surface Details That Are Often Misunderstood

A Tank Does Not Need to Be Half Full

For an ideal rectangular tank with vertical sides, the free surface moment depends mainly on the dimensions of the liquid surface, not directly on the quantity of liquid below it.

A tank can therefore produce a substantial correction while considerably less or more than half full, provided the free surface still extends across the tank breadth.

Actual values change with tank shape, internal structure and liquid level. Approved free surface tables remain the correct source for real vessels.

Several Slack Tanks Can Be Worse Than One Planned Transfer

Free surface corrections are cumulative:

Ballasting many tanks simultaneously can produce a large temporary reduction in GM even when the final condition is satisfactory.

Sequential filling or emptying procedures are often used to reduce the number of slack tanks during transfers.

Transverse Subdivision Is Highly Effective

Because tank breadth is cubed in the free-surface inertia equation, dividing a wide tank can greatly reduce its effect.

For a rectangular tank divided into two equal longitudinal compartments:

assuming both compartments have free surfaces and the division is complete.

The centreline bulkhead has not reduced the total liquid volume, but it has reduced the distance through which the liquid can move.

The Double Penalty of Consuming Low Fuel or Ballast

Consumable liquids stored in double-bottom tanks may affect stability in two ways as they are used:

1. Removing weight from low in the vessel can raise the actual KG.
2. Leaving the tank slack introduces free surface correction.

The vessel can therefore lose GM through both weight redistribution and free surface effect at the same time.

This explains why an arrival condition can be more limiting than the departure condition even though the ship is lighter.

Stability must be checked throughout the voyage, not only immediately after loading.

Complete Worked Ship Stability Example

The following example is illustrative. Its hydrostatic values are not associated with a particular vessel and must not be used as approved operational data.

General IMO Intact Stability Criteria

The following are the widely used general criteria for ships to which Part A, Chapter 2 of the 2008 Intact Stability Code applies:

These values are reproduced in current UK Maritime and Coastguard Agency instructions implementing the 2008 IS Code.

They should not be applied blindly to every craft. Different or additional requirements may apply to:

• Passenger ships
• Fishing vessels
• Timber deck cargo ships
• Grain carriers
• Offshore supply vessels
• Container ships
• High-speed craft
• Tugs
• Sailing vessels
• Pontoons and barges
• Vessels subject to damage stability requirements

Flag-state rules, classification requirements, approved stability information and vessel-specific operating limits take precedence.

Factors That Change Stability During Operation

Stability is not fixed when the vessel leaves the shipyard. It changes with every loading condition.

Raising KG

KG increases when:

• Cargo is loaded high
• Containers are added to upper tiers
• Deck cargo is increased
• Heavy equipment is installed above the original design position
• Ice accumulates on masts, rails or superstructures
• A crane lifts a suspended load
• Low fuel, ballast or stores are consumed

A higher KG reduces GM and generally reduces GZ across the curve.

Lowering KG

KG can be reduced by:

• Adding weight low in the vessel
• Removing high weight
• Relocating equipment downward
• Using appropriate low ballast tanks

Ballasting must still account for free surface during the transfer and for any effects on draft, trim, freeboard and longitudinal strength.

Changing KM

KM is not a fixed vessel constant.

It varies with:

• Draft
• Displacement
• Trim
• Waterplane shape
• Hull geometry at the operating condition

A hydrostatic KM value must be taken from the correct displacement or draft. Using KM from a neighbouring table row without interpolation can introduce a significant error when GM is already small.

Loss of Watertight Integrity

Open doors, hatches and ventilators can reduce the effective downflooding angle.

A vessel may have a satisfactory theoretical GZ curve based on closed weathertight boundaries while possessing much less usable stability when openings are left unsecured.

Freeing ports and drainage arrangements also matter. Water retained on deck adds weight high in the vessel and creates a broad free surface.

Cargo and Weight Shifts

A transverse weight shift moves G away from the centreline and creates a heeling moment:

where \(d\) is the transverse distance moved.

The equilibrium heel angle occurs where the heeling-arm curve intersects the vessel’s GZ curve.

Cargo securing is therefore part of stability control. A vessel can begin a voyage with an acceptable loading calculation and lose that condition when cargo moves.

Common Stability Calculation Errors

Using Deadweight Instead of Displacement

Righting moment and free surface correction use total displacement, not deadweight.

Deadweight excludes the lightship. Substituting deadweight can produce a major error.

Using Uncorrected GM

If slack tanks exist, the solid GM is not the operational GM.

Free surface correction must be applied according to the approved stability information.

Applying \(GZ = GM\sin\phi\) at Large Angles

This equation describes initial stability. Large-angle GZ must come from KN data, cross-curves, an approved loading computer or suitable vessel-specific analysis.

Treating Positive GM as Proof of Safety

Positive GM only confirms stable initial equilibrium. The curve may still have insufficient area, low maximum GZ or an early downflooding angle.

Using Hydrostatics From the Wrong Condition

KM, KB, displacement, LCF, MCT and other hydrostatic values change with draft and trim.

All values used in one calculation should refer to the same loading condition and datum.

Ignoring Temporary Transfer Conditions

The starting and final ballast conditions may both be acceptable while an intermediate condition contains too many slack tanks.

The transfer sequence itself must be considered.

Assuming the Largest Possible GM Is Best

Excessive GM can create severe rolling accelerations and high cargo-securing loads.

Adequate stability should be achieved without producing unnecessarily violent motion.

How the NauticalSolver Stability Calculators Fit Together

No single calculator represents the entire stability assessment. Each tool answers a different part of the problem.

Transverse Metacentric Height Calculator

Use the GM Calculator to determine:

• BM from waterplane inertia and displaced volume
• KM from KB and BM
• GM from KM and KG
• Corrected GM after free surface correction

This is the starting point for initial stability calculations. The calculator supports both direct hydrostatic input and an inertia-based route.

Free Surface Correction Calculator

Use the Free Surface Correction Calculator to:

• Sum free surface moments
• Estimate rectangular-tank FSM
• Calculate FSC from displacement
• Apply the correction to solid GM

The tank-geometry method is suitable for preliminary estimates. Approved tank tables should be used for vessel operations.

Transverse Stability Calculator

Use the Transverse Stability Calculator to combine:

• KB, BM, KM and KG
• Free surface correction
• Heel angle
• Approximate small-angle GZ
• Righting moment
• External heeling moment

It is useful for showing how a change in GM alters the vessel’s initial response.

GZ Curve Generator

Use the GZ Curve Generator to examine:

• GZ across several heel angles
• Maximum righting arm
• Angle of maximum GZ
• Range of positive stability
• Area beneath the curve
• Effect of changing KG

A generated educational curve should not replace approved KN data or a class-approved loading computer.

Trim and MCT 1 cm Calculators

Transverse stability and trim are separate calculations, but they interact through the operating condition.

A large trim change can alter:

• Draft distribution
• Waterplane geometry
• KM
• Freeboard
• Downflooding angles
• Propeller and rudder immersion

The Trim Calculator and MCT 1 cm Calculator can be used alongside the stability tools when weights are loaded, discharged or shifted longitudinally.

Final Remarks

Ship stability begins with the relationship between weight and buoyancy, but it does not end with GM.

KG describes where the vessel’s weight acts. KB and BM describe how the immersed hull and waterplane position the metacentre. Together they produce KM and GM, which determine the vessel’s initial response to heel.

The GZ curve then carries the analysis beyond the small-angle region. It shows how the righting arm grows, where it reaches its maximum, how much stability energy is available and where the vessel’s restoring ability disappears.

Free surfaces, cargo movements, fuel consumption, topweight and watertight integrity can alter this behaviour throughout a voyage.

The most reliable way to interpret a stability condition is therefore to read it in layers:

• GM for initial stability
• GZ for stability at each heel angle
• Curve area for stability reserve
• Downflooding angle for the usable limit
• Approved loading information for the actual vessel

A ship does not possess one permanent level of stability. It possesses a different stability condition for every combination of displacement, trim, KG, tank state and watertight configuration.

References and Further Reading

• International Maritime Organization, Ship Design and Stability and the International Code on Intact Stability, 2008.
• UK Maritime and Coastguard Agency, Instructions to Surveyors – Intact Stability.
• Transport Canada, Stability Study Guide for Applicants to the Fourth-Class Engineer Certificate.
• Transport Canada, Adequate Stability and Safety Guidelines for Fishing Vessels.