Published 25 April 2026 · last updated 28 April 2026

Cross Curves of Stability and KN Tables

Cross curves of stability (also called isocline curves) and KN tables are pre-computed naval architecture data sets that provide the KN value (the perpendicular distance from the keel reference K to the buoyancy force line of action at a given heel angle and displacement) as a function of displacement and heel angle. The data enables rapid generation of the GZ curve for any loaded condition by the simple formula $GZ = KN - KG \sin(\phi)$, where $KG$ is the centre of gravity height for the loaded condition. The cross curves and KN tables are the standard reference data for stability calculation, included in every ship’s trim and stability booklet, calculated by the shipyard at the design stage using the hull’s lines plan and validated by the inclining experiment that determines the lightship’s $KG$ and $LCG$ at delivery. The data is then loaded into the vessel’s IMO-approved loading computer for operational use; on every loaded condition the loading computer interpolates KN values at the actual displacement and at multiple heel angles to generate the GZ curve, applies the as-loaded $KG$ (including free surface correction), and checks the resulting GZ curve against the IMO 2008 IS Code criteria. ShipCalculators.com hosts the principal computational tools: the KN table interpolation calculator, the cross curves of stability calculator, the GZ from KN calculator, the KG calculator with free surface correction, the GZ curve calculator and the IS Code criteria check calculator. A full listing is available in the calculator catalogue.

Contents

Background

Why cross curves matter

The GZ curve is the fundamental description of a vessel’s stability behaviour. The GZ value at each heel angle depends on:

The hull form (which determines how the centre of buoyancy moves as the vessel heels).
The displacement (which determines the underwater volume and therefore the centre of buoyancy).
The centre of gravity $KG$ of the loaded condition (which determines the geometric difference between the buoyancy force line and the gravity force line).

For each loaded condition (cargo distribution, ballast, fuel state), recomputing the full GZ curve from the hull lines plan would be computationally expensive. The standard approach is to pre-compute the GZ contribution from the hull form (which depends only on displacement and heel angle, not on KG) and to apply the as-loaded KG correction at runtime.

The pre-computed data is the cross curves of stability or equivalently the KN tables, calculated by the shipyard at the design stage for a representative grid of displacements and heel angles.

Definitions

K: the keel reference point, conventionally located on the centreline at the keel (the lowest point of the hull at the midship section). All vertical measurements are referenced to K.
B: the centre of buoyancy at the heeled condition.
N: the perpendicular foot from K to the buoyancy force line of action (the vertical line through B at the heeled condition).
KN: the distance from K to N, measured perpendicular to the buoyancy force line.
KG: the height of the centre of gravity above K, calculated for the as-loaded condition.
GZ: the righting arm at the heeled condition.

The geometric relationship is:

$$ GZ = KN - KG \cdot \sin(\phi) $$

where $\phi$ is the heel angle. The relationship holds at all heel angles (small and large) and for both wall-sided and non-wall-sided hull behaviour.

Pre-computed data structure

The KN table is a 2D matrix indexed by:

Heel angle ($\phi$): typically 0, 5, 10, 15, 20, 30, 40, 50, 60, 75, 90 degrees.
Displacement ($\Delta$): typically 10 to 20 values covering the operating envelope (light ballast to full load).

Each cell of the matrix gives the KN value (in metres) at the corresponding heel angle and displacement.

The cross curves of stability are the same data plotted graphically: each curve in the family corresponds to a single heel angle, plotted as KN vs displacement.

Calculation methodology

Hull form integration

For each (heel angle, displacement) pair, the calculation proceeds as follows:

Heel the hull to the specified angle $\phi$.
Find the heeled waterline that floats the specified displacement. For “free trim” cross curves, the longitudinal trim is also adjusted to maintain longitudinal equilibrium; for “fixed trim” cross curves, the trim is fixed at zero (with subsequent corrections).
Calculate the underwater volume centroid (the centre of buoyancy B at the heeled condition), by integration over the heeled hull surface.
Calculate KN as the perpendicular distance from K to the vertical line through B in the heeled coordinate frame.

The integration is typically performed numerically, using the hull’s lines plan (or a 3D surface model) discretised into stations and waterlines. Modern naval architecture software (NAPA, AVEVA Marine, Bureau Veritas BV-LCM, DNV NAUTICUS) automates the integration.

Free trim vs fixed trim

The two principal cross curve conventions:

Free trim cross curves: the longitudinal trim at each heel angle is adjusted to maintain longitudinal equilibrium. Most accurate but more computationally intensive.
Fixed trim cross curves: the longitudinal trim is fixed (typically at zero or at the design trim). Simpler to calculate but requires trim correction at runtime.

Modern Class society practice is overwhelmingly free trim. The fixed trim convention is preserved in some legacy documentation but is being phased out.

Standard heel angle grid

The IMO IS Code criteria reference specific heel angles (notably 30 and 40 degrees, plus the angle of vanishing stability and the angle of maximum GZ). The cross curves are conventionally tabulated at:

0 degrees (trivial, $KN = 0$ for the upright condition).
5 degrees (initial GM check).
10, 15, 20 degrees (small-angle behaviour).
30 degrees (IS Code area-to-30 criterion).
40 degrees (IS Code area-to-40 criterion).
50, 60 degrees (large-angle stability).
75, 90 degrees (extreme heel).

Some Class societies require additional intermediate angles (e.g. every 5 degrees) for more detailed checks.

Standard displacement grid

The displacement grid typically covers:

The lightship displacement (no cargo, minimum ballast).
Several intermediate ballast and partial-load conditions.
The design displacement.
The maximum displacement (Summer Load Line + reserve for fresh water transition).

Ten to twenty displacement values are typical, providing sufficient resolution for accurate interpolation.

Inclining experiment

Purpose

The inclining experiment is a controlled test performed at delivery (and after major modifications) to determine the lightship’s $KG$ and $LCG$. The experiment:

The vessel is in light condition (minimal cargo, minimum ballast, all consumables at known levels).
Known weights are moved from one side of the vessel to the other.
The resulting heel angle is measured precisely.
The metacentric height $GM$ is calculated from the moment-to-heel relationship.
With $GM$ known and $KM$ available from the hydrostatic curves at the test displacement, the lightship $KG$ is calculated.

Procedure

The IMO 2008 IS Code (Resolution MSC.267(85), Annex 1, Chapter 8) specifies the inclining procedure:

Moveable weights: typically 10 to 100 t, positioned at the maximum athwart distance.
Multiple weight positions: typically 4 to 8 different positions to generate multiple heel angles.
Pendulum or U-tube: precise heel measurement.
Wind, wave, and tidal correction: weather conditions controlled to minimise external disturbances.
Free surface correction: any slack tanks identified and the free surface effect computed.
Verification: independent calculation by the surveyor.

The inclining experiment results in:

The verified lightship displacement (with applicable fluid corrections).
The verified lightship $KG$.
The verified lightship $LCG$ (longitudinal centre of gravity).

Documentation

The inclining experiment results are recorded in the Inclining Test Report, which becomes part of the trim and stability booklet. The Class society review and approve the report.

If the lightship $KG$ as determined by inclining differs significantly from the design value (typically more than 1 to 2% difference), the trim and stability booklet must be revised to use the actual values.

Use in the loading computer

Loading computer architecture

A modern loading computer comprises:

Database of cross curves / KN tables (loaded from the trim and stability booklet at vessel delivery).
Database of tank capacities and free surface moments for each fuel, ballast and cargo tank.
Lightship database (lightship displacement, $LCG$, $KG$, including any modifications).
User interface for entering cargo distribution, fuel state, ballast levels.
Calculation engine: integrates the inputs to produce displacement, draught, trim, $KG_{eff}$ (with free surface correction), GZ curve, and IS Code compliance check.
Compliance reporting: clear pass/fail status for each IS Code criterion.

Calculation flow

For each loaded condition:

Inputs: cargo distribution (per hold), ballast distribution (per tank), fuel and consumables (per tank).
Calculate displacement: sum of lightship + cargo + ballast + consumables.
Calculate trim: from the longitudinal moment balance (lightship $LCG$ + cargo and ballast moments vs longitudinal centre of buoyancy at the resulting waterline).
Calculate $KG$ as-loaded: weighted average of lightship $KG$ + cargo $KG$ + ballast $KG$ + consumables $KG$.
Calculate free surface correction $FSC$: sum over all slack tanks of (moment of inertia of free surface area × density / displacement).
Calculate $KG_{eff} = KG + FSC$.
Interpolate KN values from the table at the as-loaded displacement, for each heel angle in the grid.
Calculate GZ at each heel angle: $GZ = KN - KG_{eff} \sin(\phi)$.
Construct the GZ curve by connecting the points.
Calculate IS Code metrics: initial GM, area under curve to 30/40 degrees, $GZ_{max}$, angle of $GZ_{max}$, angle of vanishing stability.
Check against IS Code criteria: pass/fail per criterion.

The calculation typically completes in less than 1 second on modern computers.

Operational role

The loading computer is used:

Pre-loading: to plan the loading sequence ensuring no intermediate condition violates IS Code criteria or hull strength limits.
During loading: to track the actual loaded condition vs plan, with continuous IS Code check.
Pre-departure: to verify the final loaded condition satisfies all criteria.
During voyage: to track the changing condition as fuel is consumed.
Pre-arrival: to plan the cargo discharge sequence.
What-if analysis: to evaluate proposed cargo redistribution, ballast transfer, or load condition variations.

IMO requirements

The IMO IS Code (Resolution MSC.267(85)) and SOLAS Chapter II-1 require:

All vessels above 5,000 GT to carry an IMO-approved loading computer.
The computer to be capable of calculating intact stability, damage stability, and longitudinal strength.
The computer to provide clear and unambiguous compliance status.
Type approval by the Class society.

Major commercial loading computers approved by all major Class societies include:

NAPA Loading Computer: dominant in the merchant fleet (approximately 80% market share).
AVEVA Marine Loading Computer (Bureau Veritas BV-LCM).
DNV NAUTICUS Loading.
Lloyd’s Register IntelliShip.
NK PrimeShip Hull.

Special considerations

Curves of “MN” (more sensitive applications)

Some applications use the “MN” formulation (the moment about the metacentre rather than KN). This is the same information in a different form; the choice between KN and MN is essentially a convention.

Cross curves vs KN tables

The cross curves (graphical) and KN tables (numerical) contain the same information. Modern practice uses the numerical KN tables for computer calculation; the graphical cross curves are retained in the trim and stability booklet for human inspection and as a backup if the computer fails.

Damage stability cross curves

Separate cross curves are calculated for damage scenarios (specific compartments flooded). The damage stability cross curves are used in the damage stability calculation under SOLAS Chapter II-1. Each damage case has its own cross curve set.

Inclining experiment for converted vessels

When a vessel undergoes major modification (lengthening, bulbous bow retrofit, conversion from one type to another, change of cargo arrangement), the inclining experiment must be repeated to verify the new lightship $KG$. The cross curves may need to be recalculated if the hull form is significantly modified.

Sister ships

For sister ships built to the same design, the cross curves are typically identical. Each individual vessel still requires its own inclining experiment because the lightship $KG$ can vary by a few percent due to fabrication tolerances.

Implications for design and operations

Design

For newbuild design, the cross curves are calculated:

Multiple times during the design iteration (as the hull form is refined).
Once formally for the final design submission to Class.
Once again after the inclining experiment to update the trim and stability booklet with verified data.

Operations

For in-service operations, the cross curves are:

Loaded into the loading computer at vessel delivery.
Updated only after major modifications.
Periodically reviewed at Class periodic surveys to confirm continued validity.

The cross curves are essentially static; they do not change during normal vessel operation. The loaded condition variability is captured through the KG and free surface correction at runtime.

References

IMO Resolution MSC.267(85): Adoption of the International Code on Intact Stability, 2008 (2008 IS Code). International Maritime Organization, 2008.
IMO Resolution MSC.281(85): Explanatory Notes to the Adoption of the International Code on Intact Stability. International Maritime Organization, 2008.
SOLAS Chapter II-1: International Convention for the Safety of Life at Sea, 1974, as amended. International Maritime Organization, 1974 with subsequent amendments.
IACS. Common Structural Rules for Bulk Carriers and Oil Tankers (CSR BC and OT). International Association of Classification Societies, 2024 edition.
DNV. DNV Rules for Classification of Ships, Pt 5 Ch 4 Damage Stability and Intact Stability. DNV, 2024 edition.
Lloyd’s Register. Rules and Regulations for the Classification of Ships. Lloyd’s Register Group, 2024 edition.
ABS. Rules for Building and Classing Steel Vessels. American Bureau of Shipping, 2024 edition.
Lewis, E. V. (editor). Principles of Naval Architecture, Volume I: Stability and Strength. SNAME, 1988.
Tupper, E. C. Introduction to Naval Architecture. Butterworth-Heinemann, 5th edition, 2013.
Biran, A. Ship Hydrostatics and Stability. Butterworth-Heinemann, 2nd edition, 2014.
Rawson, K. J. and Tupper, E. C. Basic Ship Theory. Butterworth-Heinemann, 5th edition, 2001.