Ship · stability
Stability
Weight acts down through the centre of gravity G; buoyancy acts up through the centre of the underwater volume, B. Upright they are in line. Heel the ship and B moves towards the low side. If it moves far enough to lie outside G, the two forces form a couple that turns the ship back. The arm of that couple is the righting lever GZ.
Load the ship
The righting-lever curve
Shaded: the areas the criteria measure. Dashed: the tangent at the origin, which reaches GM at one radian (57.3°); for small heels GZ ≈ GM sin θ.
| IMO criterion (2008 IS Code, A 2.2) | Required | This loading |
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The criteria come from IMO IS Code 2008. They apply to real ships of full form whose curves come from their own hull lines; here they are applied to a prism so you can see how loading changes them, not to certify anything. Downflooding through openings, wind and the effect of the ship’s ends are not modelled.
How it is computed
Nothing here comes from a stored table. For every angle the program cuts the hull section with a waterline, moves the waterline until the cut-off area equals the upright underwater area (the displacement does not change when a ship heels), and takes the centroid of that area as B. Then GZ is the horizontal distance between B and G. The hull is a prism, the same section along the whole length, so the calculation is exact for a barge and an approximation for a ship.
Checked when the site is built: for a box, KB = T/2 and BM = B²/12T; GZ agrees with the wall-sided formula to 10⁻⁶ m up to deck-edge immersion; a circular section gives GZ = (KM − KG) sin θ at every angle, because its metacentre is fixed at the centre; the angle of loll matches tan θ = √(2|GM|/BM); and GZ(−θ) = −GZ(θ).
Liquid in a partly filled tank flows to the low side as the ship heels, so its weight acts as if G were higher. For a rectangular tank of breadth b the virtual rise is (ρtank/ρsea) · l b³ / 12 ∇. The cube makes wide tanks dangerous, which is why tanks are divided by fore-and-aft bulkheads. Try the slack-tank slider.
Older practice gave stability as a table of KM and a “stability curve” for a few loadings, worked out by hand with planimeters and integrators. The 1968 and 1993 IMO recommendations became the mandatory 2008 IS Code, which added weather criteria for wind and rolling. The physics of G, B and M has not changed since Pierre Bouguer named the metacentre in 1746.