The GZ Curve and Intact Stability Criteria

Read first: flotation, centres and metacentric height and stiff and tender vessels. This topic builds on G, B, M and GM.

What the GZ curve is

In short: the GZ curve shows the righting lever (GZ) for every angle of heel, for one loading condition. Bigger GZ = more force to bring the ship back upright.

• GZ = the righting lever: the horizontal distance between the centre of gravity G and the line of the buoyancy force through B when the ship heels.
• Righting moment = displacement × GZ (MCA General Ship Knowledge syllabus 2.2.2.3).
• The curve is drawn for one displacement and one KG. Change the loading and you get a new curve.
• Horizontal axis: angle of heel (°). Vertical axis: GZ (m).
• GM describes only small angles (initial stability). The GZ curve describes the ship’s behaviour at large angles.
• The stability booklet holds the curves of statical stability (GZ curves) for the ship’s standard loading conditions, with the lightship weight and its vertical centre of gravity.
• GZ curves must be produced for the loading conditions in which the yacht operates, to show that the criteria are met (REG Yacht Code Part A, 11.1(4)).
• On the bridge: before you lift the tender with the crane, take on fuel or fill the pool, check the loading condition against the stability booklet or the approved loading computer.

Reading the curve

In short: one curve, six things to name. Examiners want you to sketch it and label it (MCA Master Yachts Stability syllabus 2.3.1.1).

• Initial GM: draw the tangent to the curve at the origin. Go up from 57.3° (1 radian): the height where the tangent cuts that line is GM. A steep start = big GM.
• Maximum GZ and the angle at which it occurs: the biggest righting lever the ship can produce.
• Angle of deck edge immersion: roughly where the curve changes from curving up to curving down (the point of inflection). After this the waterplane shrinks and GZ grows more slowly.
• Range of positive stability: all the angles at which GZ is positive, from upright to the angle of vanishing stability.
• Angle of vanishing stability: where the curve crosses zero again. Heel past it and the ship capsizes.
• Dynamical stability: the area under the curve (m·rad). It measures the work needed to heel the ship to a given angle. This is what the surveyor checks against the criteria.
• Static vs dynamic: static stability is the lever (GZ) at one angle; dynamical stability is the energy (area) up to that angle (MCA Master Yachts Stability syllabus 2.3.1.7).
• Down-flooding angle (θf): the heel at which openings that cannot be closed weathertight go under water. The criteria stop counting area at this angle if it comes before 40° (IMO MSC.267(85) Part A 2.2.1).

How loading changes the shape

In short: raise G and the whole curve shrinks. More freeboard stretches it; more beam steepens the start.

• Stiff vessel (large GM): steep start, big GZ values. Tender vessel (small GM): flat start, small GZ values.
• Raising G (higher KG): smaller GZ at every angle, smaller maximum GZ, smaller range of stability. Lowering G does the opposite.
• More freeboard: the deck edge goes under later, so maximum GZ comes at a larger angle and the range gets bigger.
• Less freeboard: deck edge and openings immerse sooner; maximum GZ comes earlier and the range is smaller.
• More beam: bigger initial GM (steeper start), but the deck edge immerses at a smaller angle.
• Free surface acts like a rise of G, so slack tanks shrink the curve too. The criteria use GM after free surface correction.
• The MCA Master Yachts Stability syllabus (2.3.1.3–2.3.1.5) asks you to explain these changes.

Angle of loll

In short: a ship with negative GM (G above M) cannot stay upright. She heels until GZ = 0 and floats there: that angle is the angle of loll.

• Upright she is unstable. As she heels, the waterplane gets wider, M rises, and at some angle G and B line up again: GZ = 0.
• Push her further and GZ becomes positive again, so she rests at the loll angle with a small range of stability left.
• A ship at loll may flop from one side to the other: she lolls to port or starboard, never upright.
• List vs loll:
  • List = G moved off the centreline by weight (added, removed or shifted). GM is positive. She leans to one side only.
  • Loll = negative GM with G on the centreline. She leans to either side and has a long, slow, sluggish roll about the loll angle.
• Common causes: weight high up, a large free surface, ice, water trapped on deck, damaged compartments flooding.
• Danger: the ship looks stable but her reserve is very small. A gust or a wave can capsize her. Act fast.
• How to correct it (lower G):
  • first make sure it is loll, not list: no off-centre weight explains the heel;
  • remove free surface where you can; take weight off high up;
  • fill an empty low tank that is divided at the centreline, low side first, then the other side (MCA Master Yachts Stability syllabus 2.4.1.5). She may lurch at first, because filling adds free surface for a while;
  • never move weight across to the high side: she may flop over to the other side to a bigger angle.

Intact stability criteria

In short: the GZ curve of every seagoing loading condition must pass six minimum figures. Yachts: REG Yacht Code Part A, 11.2(1)(a). Ships: IMO 2008 IS Code, MSC.267(85) Part A 2.2. The figures are the same.

• Memory aid, in order: 55 – 90 – 30 (areas, thousandths of m·rad), then 0.20 at 30, max at 25+, GM 0.15.

• The 0.15 m minimum GM is sometimes credited to “the load line rules”. Quote the texts that actually state it: REG Yacht Code 11.2(1)(a)(v) and IS Code Part A 2.2.4 (“initial metacentric height GM₀ shall not be less than 0.15 m”).
• If a yacht cannot meet (i)–(v), the Administration may agree alternative but equivalent criteria (REG YC 11.2(1)(a)(vi)).

Short Range Yachts

• If a Short Range Yacht cannot meet the standard criteria, it may use REG YC 11.2(2):
  • area up to 15° ≥ 0.07 m·rad when maximum GZ is at 15°; area up to 30° ≥ 0.055 m·rad when maximum GZ is at 30° or more;
  • maximum GZ between 15° and 30°: required area = 0.055 + 0.001 × (30° − θmax) m·rad;
  • area 30°–40° (or down-flooding angle) ≥ 0.03 m·rad;
  • GZ ≥ 0.20 m at 30° or more;
  • maximum GZ at not less than 15°;
  • GM after free surface correction ≥ 0.15 m.
• REG YC 11.2(3) gives a further set of alternative criteria, with maximum GZ at not less than 20°.

The weather criterion (IS Code)

• Ships under the IS Code must also show they can survive beam wind plus rolling (IMO MSC.267(85) Part A 2.3).
• Under a steady wind, the angle of heel must not exceed 16° or 80% of the angle of deck edge immersion, whichever is less.
• After rolling to windward and a gust (gust lever = 1.5 × steady wind lever), the area of righting energy left (area b) must be at least the area of heeling energy (area a).

Damage stability

In short: the intact criteria are not the whole story. There are minimum standards for the ship after flooding too. For yachts: REG Yacht Code Part A, 11.3.

• Applies to all yachts except Short Range Yachts (for them it is recommended where practicable) (REG YC 11.3(1)).
• Not required for yachts that fully comply with the ICLL conditions of assignment (REG YC 11.3(2)).
• Minor damage is assumed anywhere along the length, but not on a watertight bulkhead; for yachts under 85 m load line length, damage is assumed not to reach any bulkhead, deck or other watertight boundary (REG YC Chapter 11 objective, 11.3(4)).
• With any one compartment flooded, the waterline must stay at least 75 mm below the weather deck, freeboard deck or bulkhead deck (REG YC 11.3(3)).
• Standard permeabilities: stores 60%; stores but not a large quantity 95%; accommodation 95%; machinery 85% (REG YC 11.3(5)).
• Residual stability after damage (REG YC 11.3(6)):
  • angle of equilibrium not more than 7°;
  • range to the down-flooding angle at least 15° beyond the equilibrium angle;
  • maximum GZ within that range at least 100 mm;
  • area under the curve at least 0.015 m·rad.
• On the bridge: keep watertight doors and hatches shut at sea. An open door turns two compartments into one and the damage standard no longer holds. See flooding and damage control and heavy weather preparation.