The GZ (righting arm) curve.

The GZ curve is a graph of the righting lever GZ (the horizontal distance between the centre of gravity G and the centre of buoyancy B, i.e. the lever arm of the righting couple) plotted against the angle of heel. It is the fundamental curve of transverse stability.

Significance: for any angle of heel, the righting moment = Delta x GZ, so the curve shows how the ship's ability to right itself varies with heel. It is obtained from the cross-curves of stability (GZ values for a fixed KG) corrected for the actual KG and free-surface effects.

Information obtained from the curve:

• The initial slope of the curve at the origin equals GM (the metacentric height), since GZ = GM sin(theta) for small angles.
• The maximum righting lever (GZ max) and the angle at which it occurs (angle of maximum stability, typically 25-40 deg).
• The range of stability: the angle from the upright to the angle of vanishing stability (where GZ returns to zero, typically 60-90 deg).
• The area under the curve up to any angle, which is proportional to the dynamical stability (the work done in heeling the ship) and is used to assess stability in a seaway and against wind heeling.
• The angle of loll (if the curve starts below the axis, indicating negative GM).
• The effect of free surface, which reduces the curve by the free-surface correction.

Role in assessing stability: the curve is compared with the statutory criteria (e.g. IMO intact stability criteria: area under the curve up to 30 deg not less than a value, area up to 40 deg, GZ max not less than 0.2 m at an angle not less than 30 deg, GM not less than 0.15 m, etc.). It is used to check the stability in all loading conditions, to determine the maximum permissible KG, and to assess the ship's behaviour in wind and waves. A ship with a large GM has a steep initial curve but a small range and a sharp, uncomfortable roll; a ship with a small GM has a shallow curve, a large range but a small righting moment. The curve therefore balances stiffness, range and dynamical stability.