GZ Righting-Arm Curve Builder (IMO IS Code)

Formula, assumptions, and limits

The righting arm at each heel angle comes from the cross-curve value KN corrected for the height of the centre of gravity:

KN - value from the ship’s cross curves of stability at heel angle phi, in metres. KN is tabulated against displacement and is independent of KG. KG_eff - effective vertical centre of gravity, the solid KG raised by the free-surface correction: KG_eff = KG + (sum of free-surface moments) / displacement. phi - heel angle in degrees.

The areas under the curve are integrated numerically by the trapezoidal rule over the tabulated heel angles, the standard approach for an arbitrary cross-curve table. The initial GM is read from the slope of the curve at the origin. The vanishing angle is the heel at which GZ returns to zero after the peak, found by linear interpolation between the bracketing tabulated points.

The results are checked against the IMO 2008 Intact Stability Code, resolution MSC.267(85), Part A general criteria for cargo and passenger ships: area 0 to 30 degrees at least 0.055 m·rad, area 0 to 40 degrees (or the downflooding angle if less) at least 0.090 m·rad, area 30 to 40 degrees at least 0.030 m·rad, GZ at 30 degrees at least 0.20 m, maximum GZ at an angle of 25 degrees or more, and initial GM at least 0.15 m.

This tool uses the KN values, displacement, and KG you enter. It does not apply the severe-wind-and-rolling (weather) criterion, grain, timber, or ship-type-specific Part B criteria. Cross-check against the approved stability booklet and the loading computer before any operational decision.

About This GZ Righting-Arm Curve Builder

This GZ righting-arm curve builder is for naval architects, ship stability officers, and marine surveyors who need the full righting-arm curve for a loading condition rather than a single GZ reading. It takes a KN cross-curve table plus a displacement, KG, and free-surface correction, then plots GZ against heel angle across the tabulated range and reports the areas under the curve, the maximum righting arm and its angle, the vanishing angle, and the initial metacentric height. It answers the question a loading computer answers on the bridge: does this condition pass the intact stability code, and by how much.

The calculator follows the IMO 2008 Intact Stability Code, resolution MSC.267(85), Part A. GZ at each heel angle is computed as KN minus the effective KG times the sine of the heel, the standard cross-curve correction, with the effective KG raised by the free-surface moment divided by displacement. Areas under the curve to 30 degrees, to 40 degrees or the downflooding angle, and between 30 and 40 degrees are integrated numerically over the tabulated ordinates. Each result is checked against the code thresholds: area 0 to 30 not less than 0.055 meter-radians, GZ at 30 degrees not less than 0.20 meters, maximum GZ at an angle of 25 degrees or more, and initial GM not less than 0.15 meters.

The tool earns its place because a single-number stability calculator cannot show reserve dynamic stability or the shape of the curve, and shape is where stability lives. The chart fills the area beneath the curve so reserve righting energy is visible at a glance, and draws a short tangent at the origin whose slope is the initial GM. A pass-or-fail strip below the curve shows every code criterion as attained versus required, so a surveyor sees not only that a condition passes but which margin is thin.

Further reading

• GZ curve and the righting arm
• Cross curves of stability and KN tables
• Intact stability

Test cases this calculator passes

Every build of this site runs these cases through the calculator above and stops the release if any figure moves outside its tolerance. Load a case to put its inputs into the calculator.

Frequently asked questions

What is the GZ righting-arm curve?

The GZ curve plots the righting arm GZ (the horizontal distance between the lines of buoyancy and gravity) against heel angle. Its shape, peak, and the area beneath it describe a ship's intact stability for a loading condition.

How is GZ computed from a KN cross-curve table?

GZ at each heel angle equals the cross-curve value KN minus the effective KG times the sine of the heel: GZ = KN - KG_eff times sin(heel). The effective KG is the solid KG raised by the free-surface moment divided by displacement.

Which intact stability criteria does this check?

The IMO 2008 IS Code, resolution MSC.267(85), Part A general criteria: area under the curve to 30 degrees at least 0.055 meter-radians, to 40 degrees at least 0.090, between 30 and 40 degrees at least 0.030, GZ at 30 degrees at least 0.20 meters, maximum GZ at 25 degrees or more, and initial GM at least 0.15 meters.