Vessel Stability

Stability is a ship's ability to return to an upright position after being inclined by an external force. Every seafarer with a deck watchkeeping certificate must understand the principles; Masters and Chief Officers must be able to verify compliance with the mandatory criteria of the IS Code 2008 before departure.

The International Code on Intact Stability, 2008 (IS Code) is made mandatory by SOLAS Chapter II-1, Part B-1, Regulation 5-1, in force for cargo ships and passenger ships on international voyages since 1 July 2010. It superseded the 1993 IMO Intact Stability Code.

Key definitions and formulae

KB — Centre of BuoyancyKB = d/2 (box-shaped vessel)
The vertical distance from the keel to the centre of the underwater volume. Buoyancy acts upward through B. For a box-shaped vessel floating at draft d, KB = d/2.
KG — Centre of GravityKG = ΣWz / W
The vertical distance from the keel to the ship's centre of gravity. KG rises when weights are added high, falls when weights are added low. Calculated by dividing the sum of (weight × height) moments by total displacement W.
KM — Transverse MetacentreKM = KB + BM; BM = I / V
The vertical distance from the keel to the transverse metacentre M. BM is the metacentric radius: I is the second moment of area of the waterplane (breadth³/12 × length for a rectangle) and V is the underwater volume. KM = KB + BM.
GM — Metacentric HeightGM = KM − KG
The distance between the centre of gravity G and the metacentre M. Positive GM (G below M) gives initial stability. GM is the initial slope of the GZ curve: GZ ≈ GM × sin θ for small angles. A larger GM gives a stiffer ship with a shorter, more violent roll period.
FSC — Free Surface CorrectionFSC = (l × b³ × ρ_liq) / (12 × W)
Free liquid surfaces in tanks generate a virtual rise in G, reducing effective GM. For a rectangular tank: FSC = (length × breadth³ × density of liquid) ÷ (12 × ship displacement W). Tanks should be kept full, empty, or divided by a centreline wash bulkhead to minimise FSC.

The GZ righting-lever curve

Righting lever GZ — At any angle of heel θ, GZ is the horizontal distance between the line of action of buoyancy (through B′) and the centre of gravity G. GZ represents the ship's ability to self-right. Initial slope of the GZ curve equals GM × sin θ (approximated as GM × θ for small angles).
Angle of vanishing stability — The angle at which GZ returns to zero after the positive peak. Beyond this angle the ship capsizes. A larger range of stability is desirable; bulk carriers and tankers typically achieve 60°–80° or more.
Static vs dynamic stability — Static stability is the GZ value at a given angle. Dynamic stability is the area under the GZ curve, measured in metre-radians (m·rad), representing the work done against heeling. The IS Code criteria are expressed in terms of area under the curve because a ship must absorb energy, not merely resist a static moment.
Roll period and stiffness — The natural roll period T = 2π × √(k² / g·GM), where k is the radius of gyration (approximately 0.35 × B for most vessels). A very large GM gives a short, stiff roll — uncomfortable and liable to cargo shifting. A very small GM gives a long, sluggish roll and small restoring moments. The stability booklet typically specifies an acceptable range of GM.

IS Code 2008 — intact stability criteria

The following six general criteria apply to cargo ships (≥ 24 m) and passenger ships. All must be satisfied simultaneously for each approved loading condition.

Criterion 1Area under GZ curve 0°–30°
≥ 0.055 m·rad
Criterion 2Area under GZ curve 0°–40° (or to flooding angle if less)
≥ 0.090 m·rad
Criterion 3Area under GZ curve 30°–40°
≥ 0.030 m·rad
Criterion 4GZ righting lever at 30° heel or greater
≥ 0.200 m
Criterion 5Angle at which GZ is maximum
Preferably > 30°, minimum 25°
Criterion 6Initial GM₀ (corrected for free surfaces)
≥ 0.150 m

Additionally, the severe wind and rolling criterion (Weather criterion, IS Code 2.3) requires that the residual area b (area available to absorb a wind gust after dynamic roll to windward) be not less than area a (area generated by steady wind to the angle of steady heel). This accounts for combined steady beam wind and rolling.

Free surface effect in practice

For a rectangular slack tank: FSC = (l × b³ × ρ_liq) ÷ (12 × W), where l = tank length, b = tank breadth, ρ_liq = density of the liquid, W = ship's displacement. Effective GM is reduced: GM_eff = GM_solid − Σ(FSC). Measures to control FSC:

Keep tanks pressed full or empty whenever practicable; a slack tank contributes maximum FSC regardless of filling level (for a given b).
Centreline wash bulkheads — dividing a tank longitudinally reduces b by half, reducing FSC by a factor of eight (b³ effect).
Sequential ballasting — avoid having many slack tanks simultaneously; complete one before opening the next.
Grain shifting boards and cargo securing — cargo shift has the same effect as a free surface and is treated analogously in special rules for grain carriers (SOLAS VI, Part C).

Worked example — box-shaped vessel

Given: Box ship 100 m × 20 m, floating at draft d = 10 m, KG = 6 m, no free surfaces.

Step 1 — KB: KB = d/2 = 10/2 = 5.00 m

Step 2 — BM: BM = b²/(12 × d) = 20²/(12 × 10) = 400/120 = 3.33 m

Step 3 — KM: KM = KB + BM = 5.00 + 3.33 = 8.33 m

Step 4 — GM: GM = KM − KG = 8.33 − 6.00 = 2.33 m

GM of 2.33 m satisfies the IS Code minimum of 0.15 m for the initial GM criterion. The full GZ curve and weather criterion would then be checked using the stability booklet or loading computer.

Damaged stability

SOLAS Chapter II-1, Part B covers damaged stability. Two approaches exist:

Deterministic (pre-2009 cargo ships) — assumes specific compartment flooding and checks residual freeboard, range of stability, and heel angles against prescriptive criteria.
Probabilistic (SOLAS 2009, Regulation 7-2 onwards) — each subdivision index s(i) is weighted by probability p(i) of flooding; attained subdivision index A = Σ p(i)·s(i) must equal or exceed required index R.
Passenger ships — governed by SOLAS Part B-1 (Reg 7-1 to 7-3) with special requirements for the Safe Return to Port concept (vessels ≥ 120 m keel laid from 2010).

Loading conditions and the stability booklet

SOLAS Regulation II-1/5-1 requires that every ship be provided with a stability booklet approved by the Administration. The booklet contains hydrostatic tables, cross-curves of stability, and an approved set of loading conditions that must each satisfy the IS Code criteria.

Light ship
Vessel complete with all permanent equipment, no cargo, no fuel, no crew or stores.
Departure — full load
100% cargo, 100% fuel and consumables, full crew and stores.
Arrival — full load
100% cargo, 10% fuel and consumables (or minimum seagoing).
Departure — ballast
No cargo, 100% fuel and consumables.
Arrival — ballast
No cargo, 10% fuel and consumables.
50% consumables (intermediate)
Cargo at departure level, 50% fuel and stores. Often the worst case for GM.

On modern vessels a type-approved loading computer (stability software) replaces manual booklet calculations for daily operations, but the approved booklet remains onboard as the reference document and must be produced for port-state control inspections.