Published 25 April 2026 · last updated 28 April 2026

GZ Curve and Righting Arm

The GZ curve, also called the curve of static stability or the righting arm curve, plots the righting arm ($GZ$, the perpendicular distance between the line of action of the buoyancy force and the line of action of the gravity force) as a function of the heel angle ($\phi$, the angle through which the vessel has rolled from the upright). The GZ curve provides a complete graphical description of a vessel’s intact stability characteristics at a given displacement and condition of loading: the initial slope at zero heel equals the metacentric height ($GM$) and characterises the vessel’s resistance to small heel angles; the maximum value ($GZ_{max}$) and the angle at which it occurs characterise the vessel’s resistance to large heel angles; the area under the curve ($\int GZ \, d\phi$) is the dynamic stability and represents the energy required to heel the vessel to a given angle; the angle of vanishing stability ($\phi_v$, the heel angle at which $GZ$ returns to zero) characterises the maximum heel angle from which the vessel can self-right. These properties are the central inputs to the IMO 2008 Intact Stability Code (IS Code) criteria adopted under SOLAS Chapter II-1, the SOLAS Chapter II-1 damage stability calculation, the operational stability assessment that the master and chief officer perform on every loaded condition (typically through an IMO-approved loading computer), the trim and stability booklet that every vessel must carry, and the damage stability calculation under SOLAS and IMO Resolution MSC.281(85). ShipCalculators.com hosts the principal computational tools: the GZ curve calculator, the GZmax calculator, the angle of vanishing stability calculator, the area under GZ curve calculator, the IS Code criteria check calculator, the GM calculator, the KN table interpolation calculator and the cross curves of stability calculator. A full listing is available in the calculator catalogue.

Contents

Background

Why GZ curves matter

The stability of a ship is not a single number but a functional relationship between heel angle and restoring effect. A vessel may have:

High initial stability (large $GM$) but poor large-angle stability (small $GZ_{max}$, small angle of vanishing stability), as for example a wide-beam ferry with a low cargo height: the vessel feels stiff at small heel but capsizes if heeled past a moderate angle.
Low initial stability (small $GM$) but good large-angle stability (large $GZ_{max}$, large angle of vanishing stability), as for example a tall slender hull with significant deck reserve buoyancy: the vessel feels tender at small heel but is hard to capsize at large angle.
Adequate stability across the full range (positive $GM$, adequate $GZ_{max}$, adequate angle of vanishing stability, adequate area under the curve), as required by the IMO IS Code.

The GZ curve is the unique tool that captures all these dimensions of stability characterisation simultaneously. Operational decisions (loading patterns, ballast plans, trim optimisation) and design decisions (hull form, cargo deck arrangement, free surface management) are made by reference to the resulting GZ curve.

Definitions

For a vessel heeled to angle $\phi$:

Centre of gravity (G): the point at which the total weight of the vessel and its cargo can be considered to act vertically downward. G is fixed in the ship for a given loading condition; it does not move with heel.
Centre of buoyancy (B): the centroid of the displaced underwater volume. B moves as the underwater hull shape changes with heel.
Metacentre (M): for small heel angles, the point about which B appears to rotate, located on the centreline above G. M is fixed for small angles and approximately constant up to about 10 degrees of heel; for larger angles M moves.
Metacentric height ($GM = KM - KG$): the vertical distance from G to M. Positive $GM$ means the vessel will return to upright; negative $GM$ means the vessel will continue to heel further.
Righting arm ($GZ$): the horizontal distance between the line of action of the buoyancy force (vertical, through B) and the line of action of the gravity force (vertical, through G), measured perpendicular to both. $GZ$ is the moment arm of the righting moment ($GZ \cdot \Delta$, where $\Delta$ is the displacement).
Heel angle ($\phi$): the angle of inclination from the upright, conventionally in degrees, positive to starboard.
Initial transverse stability: the behaviour at small heel angles, characterised by $GM$.
Large-angle stability: the behaviour at heel angles beyond approximately 10 degrees, where $GZ$ no longer follows the small-angle approximation.

Wall-sided approximation

For very small heel angles (typically less than 5 to 7 degrees), the wall-sided approximation holds:

$$ GZ \approx GM \sin(\phi) $$

This approximation assumes the underwater hull cross-section does not significantly change shape between upright and the heeled condition. The approximation breaks down at larger angles when the deck enters the water, the bilge emerges, or other significant hull-shape changes occur.

For typical merchant ship hull forms, the wall-sided approximation is reasonably accurate up to:

Bulk carriers, tankers: 7 to 12 degrees (broadly box-shaped underwater hull).
Container ships: 5 to 10 degrees.
Cruise ships: 5 to 8 degrees (high freeboard, wide beam).
Ro-ro ferries: 5 to 10 degrees.
Naval and high-performance hulls: typically much smaller (1 to 3 degrees).

Beyond the wall-sided range, the GZ value must be calculated through full hydrostatic analysis (see Methodology below).

Calculation methodology

Cross curves of stability and KN tables

The standard method for calculating GZ is through cross curves of stability (also called isocline curves) or KN tables, both pre-computed at the design stage and provided to the vessel in the trim and stability booklet.

The cross curves of stability plot the KN value as a function of displacement, for each of a series of heel angles (typically 0, 5, 10, 15, 20, 30, 40, 50, 60, 75, 90 degrees). $KN$ is the perpendicular distance from the keel reference point K (a fixed point on the centreline at the keel, used as the reference origin for vertical measurements) to the buoyancy force line of action, at a given heel angle and displacement.

The relationship between $KN$ and $GZ$ is:

$$ GZ = KN - KG \sin(\phi) $$

where $KG$ is the height of the vessel’s centre of gravity above the keel reference, calculated for the as-loaded condition.

Therefore for any loaded condition:

Calculate the displacement ($\Delta$) and as-loaded $KG$ from the hydrostatics and the disposition of weights (lightship + cargo + ballast + fuel + stores).
Adjust $KG$ for free surface effect (slack tanks raise the effective $KG$).
For each heel angle, look up $KN(\phi, \Delta)$ from the cross curves or KN table.
Compute $GZ(\phi) = KN(\phi, \Delta) - KG \sin(\phi)$.

The resulting set of $(\phi, GZ)$ pairs defines the GZ curve for the loaded condition.

Direct hydrostatic calculation

For modern computer-aided stability calculation (typically through an IMO-approved loading computer, e.g. NAPA, Aveva, Bureau Veritas BV-LCM, DNV NAUTICUS), the GZ is calculated directly from the hull form rather than through cross curves:

Hull form definition: the hull is represented as a discretised surface (typically several thousand panels).
Heeled waterline determination: for each heel angle and the as-loaded displacement, the waterline is found that floats the displacement, accounting for trim equilibrium.
Underwater volume and centroid: the underwater volume and its centroid (the centre of buoyancy B) are integrated.
GZ calculation: $GZ$ is computed as the horizontal distance between B and G, in the heeled coordinate frame.

The direct calculation is more accurate than KN-table-based interpolation, particularly for large angles and for unusual loaded conditions, and is the modern industry standard.

Effect of trim

For each heel angle, the trim equilibrium must be found: as the vessel heels, the longitudinal trim may change to maintain equilibrium of the longitudinal moment. Some KN tables assume fixed trim (typically zero trim or design trim) and apply a correction; modern direct calculations find the equilibrium trim explicitly.

Effect of free surface

Free surface effect reduces the effective metacentric height by the free surface correction (FSC), which is added to $KG$ to give an effective $KG$:

$$ KG_{eff} = KG + FSC $$

$FSC$ is calculated for each slack tank as the moment of inertia of the free surface area divided by the displacement, summed across all slack tanks.

IMO 2008 IS Code criteria

The IMO 2008 Intact Stability Code (IS Code), adopted as Resolution MSC.267(85), specifies six general criteria that the GZ curve must satisfy for any seagoing condition:

Initial $GM$: $GM_0 \ge 0.15$ m at all conditions.
Area under GZ curve to 30 degrees: $\int_0^{30°} GZ \, d\phi \ge 0.055$ m·rad.
Area under GZ curve to 40 degrees (or to downflooding angle if smaller): $\int_0^{40°} GZ \, d\phi \ge 0.090$ m·rad.
Area under GZ curve from 30 to 40 degrees: $\int_{30°}^{40°} GZ \, d\phi \ge 0.030$ m·rad.
Maximum GZ angle: the maximum $GZ$ shall occur at a heel angle of at least 25 degrees.
$GZ_{max}$ value: $GZ_{max} \ge 0.20$ m.

Plus additional criteria for specific vessel types (passenger ships, bulk carriers, tankers, ro-ro vessels, grain carriers, livestock carriers).

The criteria define the minimum acceptable GZ curve. For each loaded condition, the actual GZ curve must be calculated and checked against the criteria.

Weather criterion

The IMO 2008 IS Code also includes a weather criterion (sometimes called the severe wind and rolling criterion) that requires the vessel to withstand a specified beam wind and rolling motion without capsizing. The criterion involves comparison of the vessel’s GZ curve with a calculated wind heeling moment.

Properties of the GZ curve

Initial GM (slope at zero heel)

The initial slope of the GZ curve at zero heel equals the metacentric height $GM$:

$$ \left. \frac{dGZ}{d\phi} \right|_{\phi=0} = GM $$

A positive $GM$ means the vessel will return to upright from a small heel; a negative $GM$ means the vessel is in unstable equilibrium and will continue to heel.

The initial $GM$ characterises the vessel’s resistance to small heel angles. A high $GM$ produces a “stiff” vessel that rolls quickly with a short period; a low $GM$ produces a “tender” vessel that rolls slowly with a long period. The optimum $GM$ depends on the vessel type, the cargo, and the operating environment; typical IMO 2008 IS Code minimum is 0.15 m.

Maximum GZ ($GZ_{max}$)

The maximum value of $GZ$ on the curve occurs at a specific heel angle, typically in the range of 25 to 60 degrees depending on the hull form. Beyond this angle, the GZ value decreases as further heel reduces the effective restoring effect.

The IMO 2008 IS Code requires $GZ_{max} \ge 0.20$ m and to occur at a heel angle of at least 25 degrees.

Range of stability

The range of positive stability is the heel angle range over which $GZ > 0$. The range begins at zero heel (where $GZ = 0$ by definition) and extends to the angle of vanishing stability ($\phi_v$), where $GZ$ returns to zero.

For typical merchant ships, the range of positive stability is in the range 50 to 80 degrees. Vessels with very low freeboard (many bulk carriers at full load) have a smaller range; vessels with high freeboard (container ships, passenger ferries) have a larger range.

Angle of vanishing stability ($\phi_v$)

The angle of vanishing stability is the heel angle at which $GZ$ returns to zero on the descending side of the curve. Beyond this angle, the restoring moment becomes negative and the vessel will continue to heel until capsize.

The IMO 2008 IS Code does not specify a minimum range of stability for general cargo vessels but does specify a minimum dynamic stability (area under the curve to 40 degrees). For some vessel types (notably grain carriers, livestock carriers), additional minimum range criteria apply.

Dynamic stability (area under the curve)

The area under the GZ curve from upright to a given heel angle represents the dynamic stability at that angle, measured in m·rad. The dynamic stability is the energy required to heel the vessel from upright to the given angle, and is the relevant measure for resistance to dynamic upsetting forces (gusts, waves, sudden cargo shift).

The IMO 2008 IS Code criteria 2, 3 and 4 specify minimum dynamic stability values at 30, 40 and 30-to-40 degree intervals.

Angle of deck immersion

The angle at which the deck edge first enters the water is a characteristic of the GZ curve and typically marks the point at which the curve begins to flatten. For typical merchant vessels, the angle of deck immersion is in the range of 20 to 35 degrees, depending on freeboard.

Angle of bilge emergence

The angle at which the bilge (the curved transition between the bottom and the side) first emerges from the water also marks a transition in the GZ curve, typically at heel angles of 30 to 45 degrees.

Downflooding angle

The angle at which the lowest downflooding opening (typically a doorway, hatch, ventilator, or other opening that cannot be made watertight) immerses is the downflooding angle. Beyond the downflooding angle, water can enter the vessel through the opening, progressively reducing the displacement and stability.

The downflooding angle effectively limits the usable range of stability for operational purposes; the IMO IS Code criteria are calculated up to the smaller of 40 degrees or the downflooding angle.

GZ curve characteristics by vessel type

Bulk carriers

Bulk carriers have characteristic GZ curves with:

Initial $GM$: typically 1.5 to 4.0 m (high, due to wide beam and low VCG when laden with high-density cargo).
$GZ_{max}$: typically 0.6 to 1.5 m, occurring at 30 to 40 degrees.
Angle of vanishing stability: typically 60 to 75 degrees.

In ballast condition, the GZ characteristics change significantly: lower $GM$ (typically 1.0 to 2.5 m), reduced range, and increased sensitivity to free surface effects.

Container ships

Container ships have characteristic GZ curves with:

Initial $GM$: typically 1.0 to 3.5 m, with significant variation depending on container loading pattern (high stack height reduces $GM$).
$GZ_{max}$: typically 0.5 to 1.5 m.
Angle of vanishing stability: typically 55 to 75 degrees.

Container ships are particularly sensitive to container loading distribution (the distribution of containers between decks and across the breadth) and to free surface in fuel and ballast tanks.

Crude oil tankers

Tankers have characteristic GZ curves with:

Initial $GM$: typically 2.0 to 5.0 m (high, due to large displacement and wide beam).
$GZ_{max}$: typically 0.8 to 2.0 m.
Angle of vanishing stability: typically 65 to 80 degrees.

Tankers are particularly sensitive to free surface effect during cargo loading/discharging, when multiple tanks are slack simultaneously.

LNG carriers

LNG carriers have characteristic GZ curves shaped by the large cylindrical or membrane cargo tanks:

Initial $GM$: typically 1.5 to 3.5 m.
$GZ_{max}$: typically 0.8 to 1.8 m.
Angle of vanishing stability: typically 60 to 75 degrees.

Passenger and ro-ro ferries

Passenger and ro-ro vessels are particularly stability-sensitive due to high freeboard, complex internal arrangement, and the potential for water on deck in damage scenarios. The IS Code requires additional criteria (Stockholm Agreement for ro-pax in NW Europe; SOLAS 90 for passenger ships).

Initial $GM$: typically 0.5 to 2.5 m (the lower end is common for ro-pax vessels with significant deck cargo height).
$GZ_{max}$: typically 0.4 to 1.2 m.
Angle of vanishing stability: typically 50 to 70 degrees.

The Stockholm Agreement requires additional dynamic stability for ro-pax vessels operating in NW European waters, reflecting the elevated capsize risk in the historical Estonia and Herald of Free Enterprise incidents.

Operational use

Loading computer

Every commercial vessel above approximately 5,000 GT carries an IMO-approved loading computer (also called electronic stability instrument) that:

Accepts the operator-input cargo, ballast and consumable distribution.
Calculates the displacement, draughts, trim, $KG_{eff}$ (including free surface correction).
Generates the resulting GZ curve.
Checks the GZ curve against the IS Code criteria and any additional vessel-specific criteria.
Reports compliance status; flags non-compliance with specific IS Code criterion failures.
Supports “what-if” scenarios for cargo redistribution, ballast transfer, fuel consumption planning.

The principal commercial loading computers are NAPA Loading Computer (used on approximately 80% of large merchant vessels), Aveva Loading Computer (Bureau Veritas BV-LCM), DNV NAUTICUS, AVEVA Marine Loading, Lloyd’s Register IntelliShip, and several smaller specialist products.

Trim and stability booklet

Every commercial vessel must carry a trim and stability booklet approved by the classification society and the flag administration. The booklet contains:

Hull form data: lines plan, hydrostatic curves, Bonjean curves, cross curves of stability or KN tables.
Light ship particulars: displacement, $LCG$, $VCG = KG$ and $TCG$ (the inclining experiment results).
Cargo, ballast and tank capacity tables: detailed tables for each tank and hold.
Free surface moment tables: for each tank.
Standard loading conditions: typically 8 to 16 standard loaded conditions covering the operational envelope, with full GZ curve and IS Code compliance check for each.
Damage stability information: damage cases, residual stability after each damage scenario.
Stability calculation methodology: instructions for the operator to calculate the stability of any non-standard loading condition.

The booklet is a controlled document and must be kept onboard; it is reviewed at every periodic survey and must be re-approved if the vessel’s structure or arrangement changes (e.g. after a bulbous bow retrofit).

Operational stability assessment

Before each voyage, the master and chief officer perform an operational stability assessment that:

Computes the loaded condition based on the actual cargo, ballast, and consumables onboard.
Generates the GZ curve for the loaded condition.
Checks IS Code compliance of the GZ curve.
Reviews the trim (typically against the trim optimisation recommendation for fuel efficiency, while satisfying stability constraints).
Reviews the heading and weather for any beam-wind or rolling concerns.

Non-compliance triggers ballast or cargo redistribution before the voyage commences.

Damage stability assessment

In addition to intact stability, the vessel’s damage stability under SOLAS Chapter II-1 must be assessed. Damage stability uses a different but related GZ-curve approach, calculating the GZ curve of the vessel after specified damage scenarios (typically flooding of one or more compartments) and checking against survival criteria.

Special considerations

Bilge keel and stability

Bilge keels (longitudinal fins on the bilge of the hull) reduce roll motion through hydrodynamic damping but do not significantly affect the static GZ curve. The effect of bilge keels is captured in the roll damping coefficient rather than in the GZ curve itself.

Anti-roll tank and stability

Anti-roll tanks (passive or active fluid tanks designed to counteract roll motion) have a static stability effect similar to other slack tanks (a free surface penalty) but provide significant dynamic roll damping. The trade-off is captured in the loading computer.

Container deck loading

Container ships are particularly sensitive to container deck loading: high deck stacks raise the effective KG significantly. The IS Code criteria can be borderline at high deck stack heights, particularly in the early stages of voyages when fuel is high.

The Container Stability Initiative (an industry group including DNV, Lloyd’s Register, MSC, Maersk, CMA CGM) has developed enhanced container stability calculation methods to address container-loading-related stability incidents (e.g. MSC Zoe in 2019, Tokio Express in 2014).

Bulk cargo shift

For bulk carriers carrying ungrouted grain or other cargoes that can shift, the International Code for the Safe Carriage of Grain in Bulk (IGC Code) requires a separate cargo shift assessment that effectively augments the standard GZ analysis.

Ice accretion

In cold and Arctic operations, ice accretion on the upper structure (rigging, masts, deck cargo, superstructure) raises the effective KG and reduces stability. The IS Code includes a specific ice accretion correction for vessels operating in defined cold-weather areas. See intact stability for the corresponding stability criteria adjustments.

Container lashing failure

Loss of container lashing (typically from heavy weather rolling beyond design assumptions) can cause container loss overboard, which sudden displacement change can cause secondary stability problems. The MSC Zoe (January 2019) container loss incident in the North Sea was a notable case study.

Implications for design and operations

Design margin

Naval architects design vessels with significant margin above the IS Code minimum criteria to accommodate operational variation and unanticipated conditions. Typical design margins:

Initial $GM$: design typically targets 0.30 to 1.00 m above the 0.15 m minimum.
$GZ_{max}$: design typically targets 0.40 to 0.80 m above the 0.20 m minimum.
Area under curve: design typically targets 50 to 100% above the minimum.

The design margin reduces the risk of inadvertent IS Code non-compliance during operations.

Operational margin

Operators (masters, chief officers) further build in operational margin by:

Avoiding loading patterns that produce borderline GZ curves.
Keeping ballast tanks fully pressed (minimising free surface) where possible.
Avoiding excessive container deck stacks.
Planning ballast/cargo redistribution as fuel is consumed.
Following the trim optimisation recommendation only where it is consistent with stability constraints.

Class society scrutiny

The classification societies (DNV, Lloyd’s Register, ABS, BV, NK, KR, RINA, CCS) review the trim and stability booklet at every periodic survey and inspect the loading computer for currency. Newbuilds undergo inclining experiments to verify the lightship $KG$ and $LCG$, which are then frozen as the booklet baseline.

References

IMO Resolution MSC.267(85): Adoption of the International Code on Intact Stability, 2008 (2008 IS Code). International Maritime Organization, 2008.
IMO Resolution MSC.281(85): Explanatory Notes to the Adoption of the International Code on Intact Stability. International Maritime Organization, 2008.
SOLAS Chapter II-1: International Convention for the Safety of Life at Sea, 1974, as amended. International Maritime Organization, 1974 with subsequent amendments.
IMO Resolution MSC.36(63): Adoption of the International Code of Safety for High-Speed Craft (HSC Code). International Maritime Organization, 1994.
IACS. Common Structural Rules for Bulk Carriers and Oil Tankers (CSR BC and OT). International Association of Classification Societies, 2024 edition.
DNV. DNV Rules for Classification of Ships, Part 3 Hull. DNV, 2024 edition.
Lloyd’s Register. Rules and Regulations for the Classification of Ships, Part 3 Ship Structures. Lloyd’s Register Group, 2024 edition.
ABS. Rules for Building and Classing Steel Vessels. American Bureau of Shipping, 2024 edition.
Bureau Veritas. Rules for the Classification of Steel Ships, Part B Hull and Stability. Bureau Veritas, 2024 edition.
Lewis, E. V. (editor). Principles of Naval Architecture, Volume I: Stability and Strength. SNAME, 1988.
Tupper, E. C. Introduction to Naval Architecture. Butterworth-Heinemann, 5th edition, 2013.
Rawson, K. J. and Tupper, E. C. Basic Ship Theory. Butterworth-Heinemann, 5th edition, 2001.