ship structureThe loading screen shows positive GM, but the GZ curve thins sharply beyond thirty degrees
GZ CURVE LAB · KN − KG SIN φ
One curve carries initial slope, maximum lever and usable angular range
KN/LK values come from an FAO teaching-vessel example and KG is 1.32 m. The dataset demonstrates formula and integration; it is not another vessel's approved stability curve.| φ | KN / LK | KG sinφ | GZ |
|---|---|---|---|
| 0° | 0.000 | 0.000 | 0.000 m |
| 10° | 0.325 | 0.229 | 0.096 m |
| 20° | 0.629 | 0.451 | 0.178 m |
| 30° | 0.866 | 0.660 | 0.206 m |
| 40° | 1.054 | 0.848 | 0.206 m |
| 50° | 1.215 | 1.011 | 0.204 m |
| 60° | 1.338 | 1.143 | 0.195 m |
| 70° | 1.428 | 1.240 | 0.188 m |
GZ₃₀ = 0.866 − 1.32 × sin30° = 0.206 mAREA₀₋₄₀ ≈ 0.102 m·rad · TRAPEZOIDAL INTEGRATIONIf KG rises 0.15 m, GZ at 30° loses 0.15×sin30° = 0.075 m and falls to about 0.131 m.GZ is the horizontal separation of the vertical weight and buoyancy lines
After a ship heels, all weight can be combined as a vertical downward force through centre of gravity G. The pressure of displaced water combines as an equal upward buoyant force through the shifted centre of buoyancy Bφ. When those opposite forces are not collinear, they form a couple. Their horizontal separation is the righting arm GZ. It is not the direct distance from G to B and it is not an angle. Positive GZ means the couple at that static angle tends to restore the vessel; negative GZ means it tends to increase the inclination.
Righting moment is displacement weight multiplied by GZ. In mass notation it is displacement mass times gravitational acceleration times GZ; tables expressed in tonnes-force often leave g implicit. The NTSB technical explanation for the Lady D capsize notes that one righting-arm curve belongs to one displacement, so length conveniently separates hull-and-loading geometry from the constant force term. Two vessels may each have a 0.20-metre lever but, at different displacements, develop very different moments.
GM gives the small-angle initial slope while the GZ curve follows larger heel
At small angles and for a smoothly changing hull form, GZ is approximately GM times sinφ. Transverse metacentre M comes from initial waterplane geometry, and GM is its vertical distance above G. A larger positive GM normally makes the curve climb more steeply near the origin. As heel grows, deck-edge immersion, flare, superstructure buoyancy and opening locations alter the path of B. A single fixed-GM sine curve then ceases to represent the actual righting lever.
A positive GM therefore describes an initial static tendency, not adequate reserve at every angle. Excessive GM can also create short-period, violent rolling that burdens people, cargo and structure; a small GM may give gentler motion yet weak initial levers. The IMO intact-stability framework uses GM together with GZ values, curve areas, the position of maximum GZ and weather criteria because no initial parameter alone captures large-angle reserve or dynamic behaviour in waves.
KN cross curves separate hull-buoyancy geometry from the loaded KG
Naval-architecture data often provide KN values—written LK in some small-vessel material—for several displacements and heel angles. This is a buoyancy-geometry lever constructed from a stated reference. Combine it with the loaded KG using GZ = KN − KG sinφ. In the FAO teaching example, KN/LK at 30 degrees is 0.866 metre. With KG 1.32 metres, KG sin30 degrees is 0.660 metre and GZ is about 0.206 metre. The construction lets one family of hull cross curves serve several valid loading centres of gravity.
The selected KN must match displacement, trim and reference convention. KG must include the moments of hull, cargo, fuel, ballast, people and suspended loads. MCA stability-booklet templates provide displacement, draught, KMT, MCT and KN tables, then use GZ = KN − KG sinθ for each loading condition. Choosing the wrong displacement column, confusing a moulded baseline with underside of keel, or omitting free-surface correction can produce a smooth-looking but physically false curve.
Maximum lever, curve area and range answer different stability questions
Each curve point is the static lever at one heel angle. The peak gives maximum GZ and its angle; a later zero crossing, if present, marks vanishing positive static stability. Area under the curve, measured in metre-radians, is conventionally used as a dynamic-stability or righting-energy indicator because it measures work against the restoring moment per unit displacement weight. Integrating with degrees produces metre-degrees, which must be multiplied by π/180 before comparison with a metre-radian criterion.
The usable curve may end before its mathematical second zero. If the lower edge of an opening that cannot be closed weathertight immerses first, the downflooding angle limits intact-stability evaluation. Incoming water then changes weight, free surface and buoyant volume, so the original intact curve no longer describes the vessel. The 2008 IS Code sets minimum areas, lever values and peak-angle provisions for ships in its scope, with additional or alternative criteria for types and sizes. One extracted number is not a universal pass mark for every yacht, fishing boat and historic ship.
Raising KG and free surfaces systematically erode the curve
If KN and displacement remain approximately fixed, raising KG by ΔKG reduces GZ at each angle by ΔKG sinφ. Raise KG 0.15 metre in the teaching example and the 30-degree lever loses 0.075 metre, falling from about 0.206 to 0.131 metre. High deck cargo, icing, a suspended lift or people concentrated aloft can raise actual or effective G. Moving weight down has the opposite first-order effect, though draught, strength and other constraints still require checking.
In a partly filled tank the liquid surface stays nearly horizontal as the ship heels, so the liquid centre moves toward the low side and creates a free-surface moment. Operational methods commonly express this as a virtual rise of KG and deduct it from usable GM or GZ. Several broad slack tanks can be worse than one pressed-full tank; liquid density, tank shape and free-surface second moment enter the correction. FAO and IMO material also treats water on deck, icing, fishing gear and suspended loads as specific risks. Every curve belongs to a defined loading state, not to the ship forever.
Openings, damage, turning and waves lie beyond one intact static curve
Positive GZ identifies a static restoring direction but does not guarantee recovery from a gust, beam sea, shifting cargo or high-speed turn. External heeling moment can exceed righting moment, and initial roll energy can cross the potential-energy reserve represented by curve area. Parametric roll, pure loss of stability and surf-riding or broaching demand dynamic treatment. IMO's work on second-generation intact-stability criteria exists precisely because vessel types respond differently in waves; a calm-water curve is essential evidence, not a complete ocean model.
Casualties make the boundary tangible. The Golden Ray investigation found that incorrect ballast entries caused the loading program to overstate stability, leaving insufficient righting arm against turn-induced forces; calculated casualty curves had markedly less area than benchmark conditions. Vasa in 1628 had no modern GZ calculation. Heavy upperworks put G too high, a gust produced severe heel, and open lower gunports admitted water, converting an intact-stability problem into flooding and sinking. Historical heel trials can reveal a crank ship, but modern terminology cannot manufacture a precise curve from missing period measurements.
Questions
Continue exploring this subject
What is the difference between GZ and righting moment?
GZ is a horizontal lever measured in metres. Righting moment is displacement weight multiplied by GZ, so vessels with the same lever but different displacement produce different moments.
Are GM and GZ the same?
No. GM is the initial metacentric height and chiefly controls the small-angle slope. GZ is the righting lever at each heel angle and requires a curve for large-angle behaviour.
How is GZ calculated from KN?
At the matching displacement, trim and heel angle, take KN and subtract KG sinφ. KG comes from the complete loaded weight moments with applicable free-surface correction.
Does positive GZ mean a ship cannot capsize?
No. External and dynamic energy, area under the curve, downflooding, loading changes, damage and applicable criteria still matter. Positive GZ states only the static couple direction at that angle.
Why is GZ-curve area measured in metre-radians?
Integrating lever over rotation represents potential-energy change per unit displacement weight, and the angle must be in dimensionless radians. Multiply a metre-degree result by π/180.
Sources
Continue the research
- International Code on Intact Stability, 2008 (MSC.267(85))International Maritime Organization
- Safety practices related to small fishing vessel stability, Chapter 6Food and Agriculture Organization
- MCA Stability Information BookletUK Maritime and Coastguard Agency
- Capsizing of Lady DU.S. National Transportation Safety Board
- Ship Design and StabilityInternational Maritime Organization
- Capsizing of Golden RayU.S. National Transportation Safety Board
- Golden Ray vessel stability reportU.S. Coast Guard / NTSB docket
- The Vasa disasterVasa Museum