Compute the righting arm GZ(θ) curve from cross-curve (KN) data and KG. Plots the curve, identifies key stability parameters, and checks area limits.
KN values at fixed heel angles from your hydrostatic tables or stability booklet
| Heel θ (°) | KN (m) |
|---|
Paste two columns (angle, KN) separated by comma, tab, or semicolon. Header row is auto-skipped. Values in degrees and metres.
Type in the vessel's Vertical Centre of Gravity (KG) in metres. This is the key variable — it shifts the entire GZ curve up or down. You can find the corrected KG in the ship's loading computer output or stability booklet.
Fill the KN table with heel angles (degrees) and corresponding KN values (metres). KN values come from the ship's hydrostatic cross-curves, typically tabulated at 10° intervals from 0° to 90°. Use "Load sample" to see the expected format.
Enter Displacement to plot the Righting Moment (RM) curve. Add a Downflooding Angle to cap the area integration at the point where unprotected openings could flood, as required by IMO stability criteria.
Click Generate GZ Curve. The chart renders instantly with key metrics alongside. Export the full report card as a PNG or PDF for inclusion in stability documentation or class submissions.
The GZ curve — formally known as the righting arm curve or curve of statical stability — is the single most important graphical tool in ship stability assessment. It expresses the vessel's ability to resist and recover from heeling across a full range of inclination angles, and is a mandatory deliverable in virtually every intact stability approval process worldwide.
When a ship heels to angle θ, the transverse separation between the centre of gravity (G) and the centre of buoyancy's vertical through the metacentre creates a righting lever GZ. A positive GZ means the ship will return to upright; a negative GZ means the ship will capsize. The GZ curve traces this lever as a continuous function of heel angle.
Where KN(θ) is the distance from the keel (K) to the centre of buoyancy's horizontal projection (N) at each heel angle, obtained from the ship's cross-curves of stability. KG is the vertical height of the centre of gravity above the keel.
A typical GZ curve for a well-loaded vessel starts near zero at 0° heel, rises to a peak, and eventually returns to zero at the angle of vanishing stability. Each feature of the curve tells you something specific about the ship's behaviour:
The gradient of the GZ curve at the origin equals GM · sin(1°) ≈ GM / 57.3. A steeper initial slope indicates greater initial stability (larger GM). However, a very steep slope combined with a short range may indicate a "stiff" ship that rolls violently.
The peak value and its angle represent the strongest restoring force available. For most cargo ships, the maximum GZ should occur at or above 25° (IMO IS Code 2008 criterion). The angle at which max GZ occurs also affects cargo securing and crane operations.
This is the angular range over which GZ remains positive — from the first positive point to the angle of vanishing stability. A wider range means the ship can tolerate larger rolling amplitudes without capsizing. IMO typically requires this to extend to at least 60°, or at least 25° beyond the maximum GZ angle.
At the AVS, the righting arm GZ returns to zero. Beyond this angle, the vessel cannot generate a restoring moment and will capsize. The AVS should be distinguished from the downflooding angle — progressive flooding can cause de facto loss of stability well before the geometric AVS.
The area under the GZ curve is perhaps the most nuanced parameter. It represents the vessel's dynamic stability energy — the work done per unit displacement in righting the ship. It is integrated in both m·rad (SI) and m·deg (practical notation) and compared against minimum area thresholds set by the IMO Intact Stability (IS) Code.
Cross-curves of stability (also called KN curves or S-curves) are a set of hydrostatic data computed by the shipyard or a naval architect for a specific vessel at a fixed displacement. They tabulate the value of KN — the lever from the keel to the intersection of the vertical through the centre of buoyancy with a horizontal — at a range of heel angles (typically 0°, 10°, 20°, …, 90°).
The power of KN cross-curves is that they are independent of KG. Once the shipyard provides the KN table for a given displacement (or range of displacements), any loading officer can compute GZ for their particular KG using the simple formula:
When using this calculator, ensure the KN values you enter correspond to the same displacement as the KG you are using. If your stability booklet has KN tables at multiple displacements, interpolate to match your actual displacement.
The IMO Intact Stability (IS) Code 2008 (Resolution MSC.267(85)) specifies minimum criteria for the GZ curve that all applicable vessels must satisfy for each loading condition. The principal weather criteria from Part A (mandatory) are:
| Criterion | Parameter | Minimum Value |
|---|---|---|
| 2.1.1 | Area under GZ curve (0° to 30°) | ≥ 0.055 m·rad |
| 2.1.2 | Area under GZ curve (0° to 40°) | ≥ 0.090 m·rad |
| 2.1.3 | Area under GZ curve (30° to 40°) | ≥ 0.030 m·rad |
| 2.1.4 | Maximum GZ (righting arm) | ≥ 0.200 m at θ ≥ 25° |
| 2.1.5 | Angle of maximum GZ | ≥ 25° |
| 2.1.6 | Initial metacentric height GM₀ | ≥ 0.150 m |
Note: When a downflooding angle θf is less than 30° or 40°, the corresponding area integration limit is reduced to θf. The weather criterion (2.1.6 and associated dynamic analysis) additionally requires that the area under the GZ curve to windward equals or exceeds the area to leeward under a wind heeling lever. This calculator computes the GZ curve area to assist with these checks — always verify against your class-approved software and the applicable version of the IS Code for your vessel type.
Certain vessel types have modified or additional GZ curve requirements under the IS Code or other IMO instruments:
While GZ at a specific angle tells you the instantaneous righting lever, the area under the GZ curve tells you how much energy is stored in the stability system — the ship's capacity to absorb the kinetic energy of rolling before capsizing. This is why it is called dynamic stability.
Mathematically, the area in m·rad is:
This calculator uses the trapezoidal rule to numerically integrate the area. Only the positive portion of the GZ curve is counted — areas where GZ is negative (if any initial heel exists) are excluded.
IMO criteria are stated in m·rad (SI units), but practitioners often work in m·deg for easier reading of tabulated values. The conversion is straightforward:
If unprotected openings (vents, skylights, companionways) are present at an angle lower than the AVS, the IS Code requires that the area integration is capped at that angle. This can significantly reduce the computed area, potentially making a vessel non-compliant even if the overall GZ shape looks healthy. Ensuring all downflooding angles are identified and entered correctly is therefore critical.
Officers use the GZ curve (often generated by the loading computer) to verify that each planned loading condition satisfies flag-state and class requirements before the vessel sails. Comparing the GZ curve for different cargo plans helps optimise loading sequences and ballast distribution.
By running this calculator with slightly different KG values, officers and naval architects can determine the critical KG — the maximum allowable KG at a given displacement before stability criteria are violated. This feeds into the maximum allowable KG curves published in stability booklets.
During heavy-lift or crane operations, a suspended load effectively raises KG. The GZ curve is regenerated with the revised KG to ensure the vessel maintains adequate stability margins throughout the lift. Righting Moment (RM) curves are particularly useful for comparing against wind heeling moments.
The GZ curve is a core topic in officer cadet training, chief mate and master examinations, and marine engineering courses. This calculator provides an interactive, visual way to understand how changes in KG, cargo distribution, and ballasting affect a vessel's stability characteristics.
While this tool computes intact GZ curves, the same principles apply to damage stability. Comparing the intact GZ curve against the residual damaged stability curve helps emergency response teams understand how much stability reserve exists during flooding casualties.
Early in hull form development, designers use parametric GZ curve analysis to benchmark proposed hull forms against stability requirements before committing to detailed hydrostatic modelling. Quick KN estimates from regression formulae can be plugged into this tool for rapid concept evaluation.
The Righting Moment (RM) is simply the GZ value multiplied by the vessel's displacement:
While GZ (metres) is dimensionless of ship size and allows comparison between different vessels and loading conditions, RM (tonne-metres) is the actual physical moment acting to restore the ship. It is most useful when:
To enable the RM curve in this calculator, enter the Displacement (Δ) in tonnes and toggle Plot RM. The RM curve will be plotted on a secondary Y-axis alongside the GZ curve.