Cross Curves of Stability and KN Tables

Cross curves of stability and KN tables are the core reference data set in every ship’s stability booklet. They encode the hull’s righting-arm geometry at a grid of heel angles and displacements, computed once at the design stage, and used for every stability check the vessel will ever undergo.

The central calculation is $GZ = KN - KG \sin\phi$, where $\phi$ is the heel angle and $KG$ is the effective centre-of-gravity height for the loaded condition. KN is purely a hull-geometry quantity. KG belongs to the loaded condition. That separation is why the method works: one pre-computed table covers the infinite number of ways the ship can be loaded.

What KN is and why it is computed about the keel

When a ship heels to angle $\phi$, the underwater volume shifts to the low side and the centre of buoyancy B moves outboard. The buoyancy force acts vertically upward through B. The righting arm GZ is the perpendicular distance between that upward force and the downward gravity force acting through G.

Computing GZ directly requires knowing G, which changes every time cargo, ballast, or bunkers change. Naval architects resolve this by computing a related quantity that depends only on the hull form. They drop a perpendicular from the keel reference point K to the line of action of the buoyancy force. That perpendicular distance is KN.

K is the conventional keel reference: the centreline intersection with the keel at the midship section, taken as the origin of the ship’s vertical coordinate system. N is the foot of the perpendicular from K to the buoyancy force line. KN is therefore a pure geometric property of the hull at a given displacement and heel angle. It contains no information about where G is.

Once KN is in hand, the righting arm follows from elementary geometry:

This identity holds for all heel angles, not merely the small-angle range. It replaces an integration over the heeled hull with a single subtraction, provided KN has already been computed. The cost is paid once, at the design office; every subsequent loading condition borrows the result.

The same data set is also called the cross curves of stability or isocline curves when plotted graphically: each curve in the family traces KN against displacement at a constant heel angle, and the curves “cross” each other as displacement changes.

The KN calculation: what the shipyard computes

For each combination of displacement $\Delta$ and heel angle $\phi$, the procedure is:

1. Heel the hull geometry to $\phi$.
2. Find the heeled waterline that floats exactly the specified displacement. For free-trim cross curves, the trim is adjusted simultaneously to satisfy longitudinal equilibrium. For fixed-trim cross curves, the trim is held at a nominal value (zero or design trim).
3. Integrate the submerged volume to find its centroid, that is, the position of B at the heeled condition.
4. Drop a perpendicular from K to the vertical line through B. The length of that perpendicular is KN.

The integration in step 3 is carried out numerically using the hull’s station-offset table or 3D surface model. At each station the submerged area at the heeled waterline is computed, and Simpson’s rule (or a higher-order quadrature) integrates the stations longitudinally. Modern naval-architecture software such as NAPA or AVEVA Marine handles the entire procedure; the result is exact to the accuracy of the hull definition.

Free trim versus fixed trim is a consequential choice. Free-trim curves are more accurate because the trim at each heeled condition is what the real ship would actually take. Fixed-trim curves overestimate KN slightly when the actual loaded trim differs from the nominal value, and a trim correction must then be applied at runtime. Classification Societies and the IMO 2008 IS Code (MSC.267(85)) both express a preference for free-trim cross curves; fixed-trim tables are still accepted in some legacy booklets but are not generated for new tonnage under most Class rules.

The heel-angle grid

The minimum grid specified by IMO practice and Classification Society unified requirements covers:

Many stability booklets add 5, 15, and 25 degree columns. Loading computers can interpolate between columns, but the IS Code criteria at 30 and 40 degrees are evaluated by integrating the GZ curve between those exact angles, so having KN tabulated at the endpoints of every integration interval is important for accuracy.

The displacement grid

The displacement axis typically runs from the lightship displacement (vessel with no cargo, minimum ballast, all consumables at known departure levels) to the maximum displacement at the Summer Load Line, extended slightly to the Tropical Load Line. Ten to twenty displacement values at roughly equal intervals are common. Finer spacing near the lightship end is sometimes used because hull geometry changes faster per unit displacement at shallow draughts than at deep draughts.

The values are chosen during stability booklet preparation to bracket every loading condition the vessel is likely to encounter. A bulk carrier operating between ballast voyage and full-load departure will always fall within the tabulated range. Interpolation between adjacent displacement rows is linear in standard loading computer implementations, which introduces small errors at hull forms with strong curvature; a few Class society rules require the spacing to be fine enough that linear interpolation error stays below 0.005 m in GZ.

From KN table to GZ curve: the full operational calculation

A complete stability calculation for a loaded condition follows eight steps, all of which a loading computer performs in under a second once the input data is entered.

Step 1: Sum all weights and moments.

where $\Delta_{light}$ is the verified lightship displacement from the inclining experiment and $m_i$ covers all cargo parcels, ballast, bunkers, freshwater, and stores.

Step 2: Compute KG for the loaded condition.

where $kg_i$ is the vertical centre of mass of each weight item above K.

Step 3: Compute the free surface correction (FSC).

Every slack tank (a tank neither full nor empty) contributes a free surface moment because liquid inside can shift when the ship heels. The correction is:

where $\rho_i$ is the liquid density in tank $i$ and $i_{fs,i}$ is the second moment of area of the free surface about the tank’s longitudinal centreline axis. The IMO 2008 IS Code (Part B, Chapter 7) specifies that the free surface correction is applied as an addition to KG, not a reduction to GZ, to avoid a common sign error.

Step 4: Compute the effective KG.

Step 5: Interpolate KN at the actual displacement.

For each tabulated heel angle $\phi_j$, interpolate linearly between the two displacement rows that bracket $\Delta$:

Step 6: Compute GZ at each heel angle.

This is the fundamental formula. The calculation is repeated for every column in the KN table to produce the discrete GZ values that define the statical stability curve.

Step 7: Integrate and characterize the GZ curve.

The loading computer connects the discrete GZ points and computes:

• The area under the GZ curve from 0 to 30 degrees.
• The area from 0 to 40 degrees (or to the downflooding angle if less than 40 degrees).
• The area from 30 to 40 degrees (obtained by subtraction).
• The maximum GZ value and the angle at which it occurs.
• The angle of vanishing stability (the heel angle at which GZ returns to zero).
• The initial metacentric height $GM_0 = KM - KG_{eff}$, where KM is read from the hydrostatic curves at the waterline corresponding to $\Delta$.

Step 8: Check the IMO 2008 IS Code criteria.

The 2008 IS Code, Part A, Chapter 2 (Resolution MSC.267(85)), establishes five minimum criteria for intact stability:

A condition that fails any one of these five checks is non-compliant. The IS Code criteria check calculator on this site applies these five checks from a user-supplied GZ table.

A worked numerical example

The following example uses realistic numbers for a 15,000 DWT general cargo vessel.

Loaded condition data:

• Lightship displacement: 4,200 t; $KG_{light}$ = 6.80 m
• Cargo in hold 2: 8,500 t; $kg$ = 5.20 m
• Cargo in hold 3: 2,100 t; $kg$ = 5.40 m
• Ballast in double-bottom tank 4P: 400 t; $kg$ = 0.65 m
• Bunkers HFO: 620 t; $kg$ = 1.10 m (full tank, no free surface)
• Fresh water: 80 t; $kg$ = 8.20 m (slack, free surface moment $= 24.0$ t.m)

Total displacement $\Delta = 4200 + 8500 + 2100 + 400 + 620 + 80 = 15{,}900$ t.

KG calculation:

Numerator: $28{,}560 + 44{,}200 + 11{,}340 + 260 + 682 + 656 = 85{,}698$ t.m

Free surface correction:

Effective KG:

KN table extract at the two bracketing displacements (illustrative values):

Interpolation fraction: $(15{,}900 - 15{,}500) / (16{,}000 - 15{,}500) = 0.800$.

GZ values:

The GZ curve peaks near 35 degrees and crosses zero between 40 and 50 degrees, indicating an angle of vanishing stability in the low-to-mid 40-degree range. The IS Code area checks would be integrated over the continuous curve constructed through these points; the trapezoidal areas at this condition are consistent with compliance at 30 degrees but marginal at 40 degrees, depending on the exact GZ peak.

This is a routine calculation for a GZ curve calculator; the manual arithmetic here is for illustration only.

The KG correction for known-G conditions: the sine correction unpacked

The formula $GZ = KN - KG_{eff} \sin\phi$ can be re-derived from first principles to show why it is exact, not approximate.

When the ship heels to angle $\phi$, the vertical distance from G (the centre of gravity) to the buoyancy force line equals the righting arm GZ. In the heeled reference frame, G has moved relative to K by a horizontal component $KG_{eff} \sin\phi$. KN is the horizontal distance from K to the buoyancy force line in the same reference frame. Subtracting $KG_{eff} \sin\phi$ from KN gives the horizontal separation between G and the buoyancy force line, which is GZ by definition.

The derivation makes two assumptions: first, that G stays fixed in the ship (no shifting cargo, no running liquids other than those already accounted for in the free surface correction); second, that K stays fixed (no keel deformation). Both hold for rigid ships with properly evaluated free surface moments.

The derivation does NOT require wall-sided geometry or small angles. It is valid at 90 degrees heel as at 5 degrees, provided the KN table was computed to 90 degrees.

Role in the stability booklet

The marine stability booklet and loading computer that every vessel carries under SOLAS Chapter II-1 Regulation 5-1 presents the KN data in two complementary forms.

Graphical cross curves plot KN on the y-axis against displacement on the x-axis, with one curve per heel angle. The family of curves lets the chief officer read off KN values visually and cross-check the loading computer output for obvious errors. The curves also make the hull form behaviour visible: a vessel with pronounced deck-edge effects will show a kink in the 40-to-60-degree family.

Numerical KN tables list the same data in a matrix. Modern stability booklets print the table with displacements in rows and heel angles in columns, or the transposed layout, depending on the naval architect’s house convention. The loading computer reads the numerical table directly. SOLAS requires the booklet to state clearly which trim convention (free or fixed) was used, the reference displacement range, and the interpolation method approved for the loading computer.

The booklet is submitted to the Classification Society for approval before delivery. The society checks that the KN table corresponds to the approved hull geometry, that the inclining experiment results have been incorporated correctly, and that the loading computer replicates the approved booklet calculations within the permitted tolerance (typically 0.005 m in GZ and 0.1 deg in trim).

The inclining experiment and lightship KG

The KN table is independent of KG; but every GZ calculation requires a reliable $KG_{eff}$, and that starts with the verified lightship $KG$ from the inclining experiment.

The experiment is performed at delivery, when the vessel is in lightship condition. Known weights (typically 10 to 50 t each, placed athwartships) are moved transversely in a controlled sequence. The resulting heel is measured by pendulums, inclinometers, or U-tube manometers. The formula:

relates the weight moved $w$, its transverse travel $d$, the ship’s total displacement $W$, and the observed heel $\theta$ to the metacentric height GM. With GM established, and KM read from the hydrostatic curves at the waterline draught during the experiment, lightship $KG = KM - GM$.

The IMO 2008 IS Code (Annex 1, Chapter 7) specifies the conditions for a valid inclining: the ship must be as near lightship as possible (typically less than 2% of displacement in moveable weights), weather conditions must be calm (less than Beaufort 2), and the results must be confirmed by at least two independent runs of the pendulums or inclinometers.

If the post-inclining $KG$ differs from the design value by more than a few centimetres, the stability booklet is reissued with the corrected value. Some flag states permit a tolerance of up to 1% of KM; others require exact incorporation of the measured value. The GM calculation from inclining experiment calculator on this site applies the above formula.

The free surface effect and its interaction with KN tables

Slack tanks reduce effective stability. Liquid in a partially filled tank shifts to the low side when the ship heels, raising the effective centre of gravity. The IS Code (Part B, Chapter 7.1) requires the free surface correction to be applied to KG before the GZ calculation, not reduced from the computed GZ after the fact.

The correction is:

where $i_{fs,i}$ is the transverse second moment of area of the free surface in tank $i$ about a longitudinal axis through the tank’s centreline. For a rectangular free surface of width $b$ and length $l$, $i_{fs} = lb^3/12$.

The interaction with the KN table method is direct: $KG_{eff} = KG + FSC$ is the quantity that enters the sine correction. The free surface correction calculator on this site computes $FSC$ for a user-defined tank inventory. As tanks fill or empty during a voyage, $FSC$ changes continuously; a loading computer updated at each fuel consumption step tracks the changing $KG_{eff}$ and flags any condition where the GZ curve drops below IS Code minima.

The metacentric height calculator explains the relationship between $GM_0$, free surface, and the initial slope of the GZ curve in more detail.

Deriving the statical stability curve and the IS Code area criteria

The GZ value at each heel angle gives one point on the statical stability curve (also called the GZ curve or righting-lever curve). Connecting these points gives a continuous picture of the vessel’s resistance to capsizing as a function of heel.

The curve’s shape has three features that the IS Code criteria target:

The area under the curve from 0 to 30 degrees captures the energy required to heel the vessel to 30 degrees. The IS Code minimum of 0.055 m.rad is not arbitrary: it represents the dynamic stability reserve against a sudden squall or wave crest that imparts impulsive heel. The dynamical stability area calculator integrates this area numerically from a user-supplied GZ curve.

The area from 0 to 40 degrees (minimum 0.090 m.rad) and the area from 30 to 40 degrees (minimum 0.030 m.rad) guard against capsize scenarios where the ship is already heeled by 30 degrees (from a partial flooding or cargo shift) and then subjected to a further heeling moment. The 30-to-40-degree zone is the last line of resistance before the GZ curve typically crosses zero.

The maximum GZ at a heel of 25 degrees or more (minimum 0.200 m) ensures the peak righting arm is not moved too close to the upright. A vessel with $GZ_{max}$ at 15 degrees may have an adequate area under the curve but will capsize quickly if heeled past the peak, with no recovery margin.

The initial metacentric height $GM_0 \geq 0.150$ m is effectively a lower bound on the curve’s initial slope. $GM_0$ is also the numerator of the small-angle approximation $GZ \approx GM_0 \sin\phi$, which is adequate to about 10 to 15 degrees for normal hull forms.

The five criteria are the global minimum standard. SOLAS, Class society rules, and flag states layer additional criteria on top: the passenger ship stability standard under SOLAS Regulation II-1/8, the grain-loading stability requirements under SOLAS Chapter VI, and the naval vessel criteria under the NATO STANAG. Any of these can bind more tightly than the IS Code minimum.

Bonjean curves and their relationship to KN computation

Bonjean curves are per-station hydrostatic curves that plot submerged area and moment of area as functions of draught at each transverse section. They are the input to the numerical integration that produces KN.

In practice, Bonjean curves and cross curves are computed from the same hull geometry file. The Bonjean integration gives the upright hydrostatics (displacement, KB, BM as functions of draught). The same station data, rotated to the heel angle, gives the heeled hydrostatics needed for KN. Modern naval architecture software integrates both from one hull model, and the Bonjean output is primarily used for longitudinal bending moment calculations (hull strength and longitudinal bending) rather than as an intermediate step in KN computation.

The historical reason for the two-step approach (Bonjean curves first, then cross curves) was that Bonjean curves could be computed by hand or by early mechanical calculators, while cross curves required more iterations. That distinction is irrelevant with current software; the connection is worth knowing because stability booklets produced before the 1980s may present cross curves derived from tabulated Bonjean integrations, and the potential for arithmetic error in those manual calculations was non-trivial.

Large-angle stability and the wall-sided formula

For wall-sided hull forms (vertical topsides above the waterline), the righting arm at moderate heel angles can be expressed by the wall-sided formula:

This is a more accurate approximation than $GZ \approx GM \sin\phi$ and is valid to about 25 to 30 degrees for most monohull forms. It is useful for hand-checking loading computer outputs: if the computed GZ at 25 degrees deviates more than about 10% from the wall-sided approximation, either the hull form is strongly non-wall-sided in that range (which is plausible for a full-form bulker with pronounced flare), or there is an input error.

The formula also shows why $GZ = KN - KG_{eff} \sin\phi$ and the wall-sided formula converge at small angles: as $\phi \to 0$, $\tan^2\phi \to 0$, the wall-sided formula gives $GZ \to GM \sin\phi$, and the KN formula gives $GZ \approx (KN/\sin\phi - KG_{eff}) \sin\phi = (KM - KG_{eff}) \sin\phi = GM_0 \sin\phi$, using the fact that $KN/\sin\phi \to KM$ as $\phi \to 0$.

Second worked example: ballast condition with significant free surface correction

The following example uses the same 15,000 DWT general cargo vessel in a heavy-ballast departure condition. The displacement is substantially lower and the free surface correction is large enough to materially affect IS Code compliance, which is the situation most likely to produce a marginal stability result in practice.

Ballast departure condition data:

• Lightship displacement: 4,200 t; $KG_{light}$ = 6.80 m
• Water ballast, double-bottom tanks 1P and 1S combined: 1,800 t; $kg$ = 0.70 m (full, no free surface)
• Water ballast, forepeak: 320 t; $kg$ = 1.20 m (full, no free surface)
• Water ballast, aft peak: 280 t; $kg$ = 1.40 m (full, no free surface)
• Bunkers HFO in settling tank: 160 t; $kg$ = 1.90 m (slack; free surface second moment $= 38.4$ t.m, density 0.98 t/m3)
• Bunkers HFO in service tank: 40 t; $kg$ = 2.10 m (slack; free surface second moment $= 9.6$ t.m, density 0.98 t/m3)
• Fresh water: 80 t; $kg$ = 8.20 m (slack; free surface second moment $= 24.0$ t.m, density 1.00 t/m3)
• Stores and crew: 20 t; $kg$ = 7.50 m (fixed)

Total displacement:

KG calculation:

Numerator: $28{,}560 + 1{,}260 + 384 + 392 + 304 + 84 + 656 + 150 = 31{,}790$ t.m

Free surface correction: three slack tanks contribute. The IS Code (Part B, Chapter 7.1) requires the actual density of each liquid to be used:

Effective KG:

KN table extract at the two bracketing displacements (illustrative values for this hull form at the lower ballast range):

Interpolation fraction: $(6{,}900 - 6{,}500) / (7{,}000 - 6{,}500) = 0.800$.

GZ calculation at each tabulated angle:

The GZ curve peaks near 30 to 35 degrees. Vanishing stability falls between 50 and 60 degrees.

IS Code 2008 (MSC.267(85)) general criteria check:

The areas are integrated by the trapezoidal rule across the discrete GZ values above. Using the values at the tabulated angles:

Area from 0 to 30 degrees, converting to radians ($30° = 0.5236$ rad):

This exceeds the 0.055 m.rad minimum by a factor of 5.1. The ballast condition, with its lower KG and wider effective beam at the waterplane, produces a substantially larger area than the loaded condition.

Area from 0 to 40 degrees:

This exceeds the 0.090 m.rad minimum.

Area from 30 to 40 degrees:

Exceeds the 0.030 m.rad minimum.

GZmax: the peak occurs between 30 and 40 degrees. By linear interpolation it is approximately $0.932$ m at 30 degrees, declining to $0.712$ m at 40 degrees, so the actual maximum lies at or just past 30 degrees. The criterion requires GZmax at 25 degrees or more, which is satisfied since the curve is still rising at 20 degrees and peaks after 25 degrees.

Initial metacentric height: at 6,900 t displacement, reading KM from the hydrostatic curves gives KM = 8.20 m (illustrative for this hull). Then:

This exceeds the 0.150 m minimum by a wide margin, as is typical for a deep double-bottom ballast condition.

All five IS Code criteria are met comfortably. The comparison with the full-load condition in the earlier example (where the area-to-40-degrees was marginal) illustrates a pattern common to general cargo vessels: ballast conditions often have large GM values and generous GZ areas because the heavy double-bottom ballast lowers G while the narrow waterplane at the ballast draught limits KM only modestly. The genuine risk in ballast for many vessels is excessive initial stiffness (very high GM), which produces a short, violent roll period rather than inadequate GZ area.

How hydrostatic software computes cross curves today

Cross curves were historically produced by hand using the ship’s lines plan and Bonjean integrations at each heel angle, a process that took draughtsmen several days for a single vessel. Modern practice uses naval-architecture software that automates the entire computation from the hull surface model.

The integration approach is the same in principle: for each combination of displacement $\Delta$ and heel angle $\phi$, the software heels the hull, finds the equilibrium waterline that satisfies exactly the target displacement (with longitudinal trim adjusted simultaneously for free-trim curves), and then integrates the moment of the submerged volume about the keel to obtain KN. The numerical difference is that the integration is now over a triangulated surface mesh rather than a set of tabulated station offsets, and the quadrature is accurate to better than 0.001 m in KN across the full heel range.

The assumed-KG convention. A subtlety in cross-curve generation is that the software needs to define the mass model to find the equilibrium trim. Some programs assume KG = 0 (all mass at the keel) and a free-trim condition, then solve for the waterline position. Others assume the vessel trims freely about the centre of flotation at each heel. The resulting KN values are not sensitive to the assumed KG because KN is a geometric property of the hull form, not the loading; the assumed KG only affects how the trim is handled at each step. What matters is that the stability booklet clearly states the trim assumption (fixed or free trim, and the nominal trim value if fixed), so the loading computer applies the correct trim correction at runtime. IMO MSC/Circ.1022 (guidelines on the KN method) recommends free-trim cross curves for all new tonnage precisely to eliminate the trim-correction step.

Verification against the inclining experiment. After the inclining experiment establishes the lightship KG, the naval architect runs the loading computer through several standard conditions and compares the GZ output against hand calculations using the approved booklet. Class rules (for example, IACS UR S1.7) specify that the loading computer must reproduce the approved condition GZ values within 0.005 m at each tabulated angle and the KM value within 0.01 m. Any deviation larger than this requires the software to be corrected before the booklet is approved.

The GZ curve calculator replicates the interpolation and GZ computation step in browser for users who want to check a specific condition without a full loading computer.

Deck-edge immersion, beam, freeboard, and superstructure effects on cross curves

The shape of the KN curve at large heel angles is controlled by hull geometry in a way that is not visible from the displacement and KG alone. Three geometric parameters dominate.

Beam. A wide vessel develops buoyancy outboard more rapidly as it heels than a narrow one of the same displacement. At small angles this shows up as a higher KM (larger BM from wider waterplane), giving higher initial GM. At large angles a wider beam sustains a larger KN because the underwater volume remains well clear of the centreline longer. This is why beam is the primary variable in resistance to capsize at large heel, and why stability regulations for fishing vessels (MSC.267(85) Part B, Chapter 5) impose minimum freeboard-to-beam ratios rather than just absolute GZ thresholds.

Freeboard and deck-edge immersion. When the low-side deck edge enters the water, the rate of change of KN with heel angle drops sharply. The deck adds no new buoyancy volume once submerged (if it is open or lightly built), so the KN curve flattens or even turns downward. On the GZ curve this appears as a change of slope: GZ rises steeply at small angles while the side is above water, then levels off or declines once the deck floods. A high-freeboard vessel like a container ship keeps the deck edge clear to 25 to 35 degrees, sustaining a rising GZ curve through the IS Code reference angles. A low-freeboard flush-deck vessel may show deck-edge immersion by 15 to 20 degrees, and its GZ peak may occur before 25 degrees, directly failing the IS Code 2.2.1(d) criterion that GZmax must occur at 25 degrees or more. The intact stability article covers how the IS Code addresses vessel types with inherently low freeboard, including open top containers and single-deck cargo vessels.

Superstructure and buoyant volume above the main deck. Enclosed superstructures contribute buoyancy once the hull heels far enough to bring them near the waterline on the low side. This effect raises KN at large heel angles beyond what the hull alone would provide, shifting GZmax to a higher angle and extending the range of positive GZ toward larger vanishing-stability angles. Classification Society rules and the IS Code Explanatory Notes (MSC.281(85)) specify under what conditions superstructure volume can be credited in the cross-curve computation: the spaces must be watertight (or weathertight, for a reduced credit), and the permeability of the space must be applied if it contains cargo, ballast, or machinery. Passenger ship superstructures typically receive full credit because the spaces are sealed and the watertight integrity is verified at each Safety Survey.

The interaction of these three parameters means that two vessels with identical displacement, KG, and GM values can have very different GZ curves at large angles. Beam, freeboard, and superstructure volume determine the shape of the curve at 40 to 60 degrees; the IS Code five-number summary (the two area figures, GZmax, its angle, and GM0) characterises the curve adequately for a pass/fail check but does not capture all the information the naval architect needs when designing a vessel’s form. The GZ curve and righting arm article discusses the curve’s shape characteristics in more detail, including the distinction between “stiff” and “tender” vessels and why a very high GM is not always a stability advantage.

Damage stability cross curves

The intact-stability KN table covers the undamaged hull. Damage stability (the vessel’s ability to survive flooding of one or more compartments) requires its own cross curve sets, one per damage case.

Under SOLAS Chapter II-1 Regulation 7, damage stability is evaluated by the probabilistic method. Each defined damage case changes the hull’s waterplane, its centre of buoyancy, and its effective freeboard. The equivalent of KN for a damage condition is computed by integrating the residual buoyancy of the flooded hull at each heel angle. The data is presented in the damage stability booklet as residual GZ curves for each damage case, not as raw KN tables (because the damage cases are too numerous for the table approach to be practical at the booklet level).

The damage stability article covers the probabilistic damage stability method and its relationship to the subdivision index S in more detail.

Limitations of the KN table method

Linear interpolation error. The standard method interpolates linearly between displacement rows. For hulls with strong flare or pronounced deadrise, the KN-versus-displacement curve is not linear, and linear interpolation between rows spaced 500 t apart can introduce errors of up to 0.010 to 0.015 m in GZ at the midpoint between rows. Some Class society rules require the naval architect to verify that the row spacing is fine enough to keep interpolation error below 0.005 m.

Trim correction when using fixed-trim curves. If the stability booklet was prepared with fixed-trim cross curves, a correction is needed when the actual loaded trim differs from the nominal. The trim correction can be several centimetres for vessels with significant trim-to-KN sensitivity, and applying it requires an additional set of trim-correction tables that many older loading computers do not implement correctly.

The GZ formula assumes no mass migration. The formula $GZ = KN - KG_{eff} \sin\phi$ assumes that the centre of gravity is fixed in the ship as it heels. Bulk cargoes that shift, liquids in improperly surveyed tanks, or hanging cargo can invalidate this assumption. The SOLAS Chapter VI grain stability requirements exist precisely because grain shift moves G and the standard GZ formula no longer applies.

KN tables are snapshot data. The tables are computed for the as-delivered hull form. Structural repairs, added top-hamper, hull-form modifications, or corrosion-related changes in shell plating can alter the hull form enough to invalidate the tabulated KN values. Class society survey requirements at each annual or five-year survey include a check that the stability booklet remains valid for the current hull form.

Free surface correction assumes small heel. The formula $i_{fs} = lb^3/12$ for a rectangular free surface is exact for upright conditions. As the ship heels, liquid in the tank shifts and the effective second moment of area changes. For small heel angles this correction to the correction is negligible; for tanks with breadths exceeding about one-third of the ship’s breadth at the deck, the error at 30 degrees can be 5 to 10% of the nominal FSC. IMO Resolution MSC.281(85) (the Explanatory Notes to the IS Code) permits the simplified constant-FSC approach in most operational contexts, noting that the over-estimation of FSC from the rectangular formula provides a conservative safety margin.

Cross curves do not capture loss of waterplane area at large heel. When the deck edge submerges and the waterplane shrinks, KN can decrease sharply in the 50-to-70-degree range for vessels with high freeboard. The KN table captures this effect only if the hull model was defined accurately to the deck edge and above, and if the integration was carried out with that geometry correctly accounted for. Errors in the upper hull definition are among the most common causes of discrepancies between loading computer outputs and incline-test results.

Practical checks and common errors

Checking sign consistency. $GZ = KN - KG_{eff} \sin\phi$ should be positive at angles below the vanishing stability angle and negative above it. If the computed GZ is positive at all angles including 90 degrees, $KG_{eff}$ is unrealistically low or the KN table contains an error.

Checking the 10-degree GZ against the small-angle approximation. At 10 degrees, $\sin(10°) = 0.1736$ and the wall-sided formula gives $GZ_{10} \approx GM_0 \times 0.174$. If the loading computer reports a GZ at 10 degrees more than 20% above this estimate, the $KG_{eff}$ or the interpolated KN value warrants re-checking.

Checking area ratios. The IS Code requires the area from 30 to 40 degrees to be at least 0.030 m.rad. For a vessel that just meets the 0.055 m.rad area-to-30-degrees criterion, the remaining area from 30 to 40 degrees should be on the order of 0.010 to 0.040 m.rad. Computed areas dramatically outside this range often indicate an integration error in the loading computer.

Displacement rounding. Loading computers sometimes round the input displacement to the nearest tabulated row rather than interpolating. A 2% rounding error in displacement can produce a 1 to 3% error in KN, which translates to 0.003 to 0.010 m error in GZ. This is enough to fail a tight IS Code check. The approval criteria for loading computers under MSC/Circ.1229 require that interpolation (not rounding) be used.

The GM from inclining experiment calculator can be used to cross-check a vessel’s current GM against the loading computer output. The intact stability article surveys the full regulatory framework, including IS Code Part B special requirements for specific vessel types.

See also

• GZ curve and righting arm covers the statical stability curve in detail, including the Pappus-Guldinus construction and the relationship between GZ shape and dynamic stability.
• Metacentric height explains $GM_0$, the initial metacentre, and the initial-stability branch of the GZ curve.
• Marine stability booklet and loading computer covers the documentary requirements under SOLAS and the type-approval process for loading computers.
• Free surface effect covers the FSC calculation, tank-by-tank treatment, and the assumptions behind the simplified formula.
• Intact stability covers the full 2008 IS Code framework including Part B special requirements.
• Hydrostatics and Bonjean curves covers the per-station hydrostatic data that feeds KN integration.
• Damage stability covers residual-GZ and probabilistic damage criteria.
• GZ curve calculator computes GZ at each heel angle from a KN table and a user-supplied $KG_{eff}$.
• IS Code criteria check calculator applies all five IS Code 2.2 criteria to a user-supplied GZ table.
• Free surface correction calculator computes the FSC from a user-defined tank list.
• GM calculator computes $GM_0 = KM - KG$ from hydrostatic data.
• GM from inclining experiment derives lightship $KG$ from inclining test measurements.
• Dynamical stability area calculator integrates the area under a GZ curve for the 0-to-30 and 0-to-40 degree zones.