Published 24 April 2026 · last updated 28 April 2026

Hydrostatics and Bonjean curves

Hydrostatics is the branch of naval architecture concerned with the buoyancy, stability, and form properties of a ship hull at rest in still water. It provides the tabulated relationship between draft and every hull-form quantity that governs whether a vessel floats safely, trims within limits, and meets the intact and damage stability criteria required by the International Maritime Organization. Bonjean curves are a companion tool, plotting the immersed cross-sectional area at each transverse station as a function of draft, so that displacement and the longitudinal centre of buoyancy can be evaluated at any waterline, including a trimmed, inclined, or damage-flooded waterline. Together, the hydrostatic table and the Bonjean curve set form the foundation of the loading manual, the stability booklet, and the structural design calculations that every IMO-regulated vessel is required to carry. ShipCalculators.com provides a dedicated suite of calculators covering displacement, waterplane integration, Tpc, MCT1cm, GM, GZ, KN curves, and the numerical methods used to derive them.

Contents

Historical development

Origins of systematic hydrostatics

The systematic mathematical treatment of floating bodies began with Archimedes of Syracuse, whose two-book treatise On Floating Bodies (ca. 250 BCE) established both the fundamental buoyancy principle and the conditions of stability for a paraboloid of revolution. His work remained largely isolated until sixteenth-century Europe revived mathematical mechanics. The Dutch engineer Simon Stevin, writing in 1586, and Evangelista Torricelli (1644) extended hydrostatic reasoning, but practical application to ship design required the separate development of integral calculus.

The first systematic application of integration to ship hull geometry was made by the Swedish naval architect and mathematician Fredrik Henrik af Chapman, whose Architectura Navalis Mercatoria (1768) contained 250 lines drawings and introduced geometric hull analysis to naval architecture. Chapman did not yet use calculus explicitly, but his table of ordinate-based calculations anticipates Simpson’s rule applied to ship sections. The French naval school - particularly Pierre Bouguer, whose Traité du navire (1746) introduced the metacentre and the metacentric height formula - provided the theoretical stability framework that hydrostatic tables would later quantify numerically.

The derivation of the metacentric radius formula BMT = IT ÷ ∇ is attributed to Leonhard Euler (1749) working independently of Bouguer. The two derivations converged on the same result, establishing the central result of hydrostatic theory.

Antoine-Charles Bonjean, a French naval engineer, published his curve method in 1844, providing the first practical tool for evaluating hull hydrostatics at arbitrary trimmed or inclined waterlines without manual re-integration of every section. The method was rapidly adopted by naval constructors in France, Britain, and the United States and remained the standard manual hydrostatic tool for more than a century.

Thomas Simpson (1710–1761) had published his quadrature rules in 1743, well before their widespread adoption in naval architecture. British naval constructors began applying Simpson’s rule to waterplane and volume calculations in the early nineteenth century, and by the 1850s it was codified in British and French naval architecture texts. The method appeared in Rankine’s Shipbuilding: Theoretical and Practical (1866) and was subsequently standardised in the curriculum of the School of Naval Architecture, London, and the Glasgow faculty.

Twentieth-century standardisation

The transition from manual to machine computation began in the 1920s with mechanical integrators such as the Amsler polar planimeter, which allowed direct measurement of areas and moments from drawn curves without manual ordinate tabulation. By the 1940s electromechanical desk calculators allowed a full hydrostatic table for a 150 m vessel to be computed in a working day rather than several weeks.

Digital computers, introduced to shipyard design offices in the 1960s, removed the computational barrier entirely. By the 1970s commercial programs such as AUTOSHIP (later evolving into Maxsurf) and NAPA were generating hydrostatic tables, cross curves, and inclining experiment predictions automatically from NURBS hull geometry. The IMO Intact Stability Code (originally Resolution A.167 of 1968, revised through MSC.267(85) in 2008) codified the criteria that these calculations must satisfy, creating a direct regulatory demand for accurate hydrostatic computation.

Archimedes’ principle and the displacement equation

Physical basis

The fundamental law governing all ship hydrostatics is Archimedes’ principle, stated ca. 250 BCE: a body immersed in a fluid experiences an upward buoyant force equal to the weight of fluid it displaces. For a floating vessel in equilibrium, buoyancy exactly equals weight. The ship therefore sinks until the displaced volume of water has a mass equal to the total mass of the ship.

The relationship is expressed as Δ = ρ × ∇, where Δ is displacement in metric tonnes, ρ is the mass density of the water in tonnes per cubic metre, and ∇ is the submerged volume in cubic metres. Standard values for ρ are 1.025 t/m³ for open ocean salt water and 1.000 t/m³ for fresh water. Dock water, river water, and canal water typically fall between those limits, commonly near 1.010–1.020 t/m³, and the actual value must be measured with a hydrometer during cargo operations, particularly when checking against the summer load line. A vessel displaced to 20,000 t in salt water (ρ = 1.025 t/m³) displaces a volume of 20,000 ÷ 1.025 = 19,512 m³. In fresh water, the same mass floats at a volume of exactly 20,000 m³, which is deeper by an amount equal to the fresh water allowance (FWA).

The fresh water allowance

The FWA in millimetres is Δ ÷ (4 × Tpc), where Δ is the summer salt-water displacement in tonnes and Tpc is the tonnes per centimetre immersion at the summer load waterline. The dock water allowance (DWA) scales this linearly with the density departure from 1.025 t/m³. These corrections are pre-computed and printed on the load line certificate for each vessel, and they govern how deep the vessel may load in ports on different waterways. The mathematical link between FWA, Δ, and Tpc is fundamental to the load line system.

The hydrostatic table

Role and production

The hydrostatic table, sometimes called the hydrostatic curves when plotted graphically, tabulates the principal buoyancy and form properties of the hull at a series of waterlines from the keel to the maximum operating draft. For a modern vessel it typically covers drafts in 25 cm or 50 cm increments. Each row represents the condition with the vessel floating at the stated draft on an even keel in water of specified density, usually 1.025 t/m³, with the designer then providing corrections or separate even-keel and trimmed tables for fresh water or light conditions.

The table is computed by the shipyard’s design office, usually with one of the commercial hydrostatics packages described later in this article. It is incorporated into the approved stability booklet and the loading manual, both of which are required to be carried on board under SOLAS chapter II-1 and the IMO Intact Stability (IS) Code.

Displacement and volume

The two most basic entries are ∇ (displaced volume, m³) and Δ (displacement, t). They are related by Δ = ρ × ∇ and increase monotonically with draft. For a vessel of 200 m length between perpendiculars, a 10 cm change in mean draft typically changes displacement by roughly 20–40 t on a small coaster and by several hundred tonnes on a large bulk carrier or tanker. The exact rate is captured by Tpc.

Tonnes per centimetre immersion

Tpc is the mass, in tonnes, required to change the mean draft of the vessel by one centimetre when the vessel is floating freely in water of density 1.025 t/m³. It is given by Tpc = Aw × 1.025 ÷ 100, where Aw is the waterplane area in square metres at the relevant draft. For a vessel whose waterplane has an area of 3,000 m², Tpc = 3,000 × 1.025 ÷ 100 = 30.75 t/cm. Tpc increases with draft on almost all hull forms as the waterplane area increases, reaching a maximum near the design load waterline. The Tpc calculator and the TPC from waterplane Simpson’s rule calculator compute this quantity directly.

Moment to change trim one centimetre

The moment to change trim one centimetre, abbreviated MCT1cm (also written as MCTC), is the trimming moment in tonne-metres required to alter the trim of the vessel by one centimetre. It is defined as MCT1cm = Δ × GML ÷ (100 × Lbp), where GML is the longitudinal metacentric height in metres and Lbp is the length between perpendiculars in metres. Because GML for a normal ship is large (typically several times the ship length), MCT1cm is substantially larger than GMT and varies slowly with draft. A change in the position of any cargo item alters trim by (mass shift × longitudinal distance) ÷ MCT1cm cm. The MCT1cm calculator performs this calculation. The trim from loading centroid calculator and the trim moment calculator extend it to full loading problems.

Centre of buoyancy

The centre of buoyancy B is the centroid of the submerged volume. Its vertical height above the keel, KB, rises with draft in a nearly linear fashion on most commercial hull forms, at a rate of approximately 0.535 to 0.560 times the draft for a box-shaped barge and somewhat differently for a ship with flared sides. KB appears in the metacentric height formula: GMT = KB + BMT − KG, where KG is the height of the centre of gravity above the keel. An accurate value of KB is therefore required to determine the initial stability of the vessel at any loading condition.

The longitudinal centre of buoyancy LCB is the fore-and-aft position of B, measured from the aft perpendicular or from amidships depending on the shipyard convention. At the design trim, LCB and LCG (the centre of gravity) must coincide; any separation produces a trimming moment of Δ × (LCG − LCB), which is absorbed by the change in waterplane geometry as the vessel trims. LCB moves forward in most hull forms as draft increases, reflecting the fuller form of the lower body.

Metacentric radii

The transverse metacentric radius BMT is the distance from the centre of buoyancy to the transverse metacentre MT. It is given by BMT = IT ÷ ∇, where IT is the second moment of the waterplane area about its centreline. For a rectangular waterplane of breadth B and length L, IT = L × B³ ÷ 12. Real waterplanes are narrower at bow and stern, so an equivalent formula using the waterplane coefficient Cwp must be applied or numerical integration must be used. The BM from waterline inertia calculator computes BMT directly from the waterplane geometry. As draft increases, ∇ grows faster than IT on ships with fine underwater bodies, so BMT generally decreases with increasing draft - this is one reason tankers and bulk carriers have a sharply falling KM curve at deep drafts and can approach stability limit conditions when fully laden.

The longitudinal metacentric radius BML = IL ÷ ∇, where IL is the second moment of the waterplane about its transverse axis through the centre of flotation. BML is much larger than BMT for all practical hull forms, typically 50–300 times the ship’s length for normal proportions, which is why trim calculations are conveniently described through MCT1cm rather than through the analogous treatment of transverse stability.

Centre of flotation

The longitudinal centre of flotation LCF is the centroid of the waterplane area in the longitudinal direction. It is the point about which the vessel pivots when a trimming moment is applied: if a weight is added exactly at LCF, the vessel sinks bodily with no change in trim. LCF is typically located near 45–52% of Lbp from aft for full-form vessels and somewhat further aft for fine-form vessels. Its position shifts as draft increases and influences how trim corrections and draft surveys are applied.

Waterplane and form coefficients

Three form coefficients appear routinely in the hydrostatic table.

The waterplane area coefficient Cwp = Aw ÷ (L × B), where Aw is waterplane area, L is waterplane length, and B is maximum beam at the waterline. Typical values range from about 0.65 for a fine fast vessel to 0.90 for a tanker at full load.

The block coefficient Cb = ∇ ÷ (Lpp × B × T), where T is the mean draft. It represents the ratio of the displaced volume to the bounding rectangular block. Full-form vessels such as ULCCs (ultra-large crude carriers) achieve Cb values above 0.85; fine-form vessels such as container ships typically fall in the range 0.60–0.72. The block coefficient calculator and the structural cross-check block coefficient calculator both compute Cb.

The prismatic coefficient Cp = ∇ ÷ (Am × Lpp), where Am is the area of the largest (midship) transverse section. It describes how the underwater body tapers toward the ends relative to its maximum cross-section and is directly related to wave-making resistance. Values from 0.55 for fast naval vessels to 0.85 for slow bulk carriers span the practical design range. The prismatic coefficient calculator computes Cp from input geometry.

The midship section coefficient Cm = Am ÷ (B × T) measures the fullness of the largest transverse section. A rectangular cross-section would have Cm = 1.00; practical values for commercial ships range from about 0.93 to 0.995. The three coefficients are linked by Cb = Cp × Cm × Cwp (with care over which length definitions are used), providing a consistency check on tabulated values.

Wetted surface area

The wetted surface area S, in square metres, is the area of the hull in contact with water. It does not appear in hydrostatics in the same sense as the form coefficients, but it is listed in the hydrostatic table because it governs frictional resistance, antifouling paint consumption, and heat transfer calculations. Classical approximate formulae, such as Denny-Mumford (S = L × (1.7T + Cb × B)) and Taylor (S ≈ C × √(Δ × L)), are still used at early design stages. The wetted surface Taylor estimate calculator and the Mumford formula calculator provide these estimates.

Numerical integration methods

Role of integration in hydrostatics

The hull of a merchant vessel is not a simple geometric solid, so buoyancy integrals cannot be evaluated analytically. Instead, the hull lines plan divides the hull into a series of transverse stations - typically 10, 20, or 21 stations numbered 0 (aft) to 10 or 20 (forward) - and numerical quadrature is applied to the offsets (half-breadths) measured at each station and waterline.

Simpson’s first rule

Simpson’s first rule, also called the three-ordinate rule or the 1-4-1 rule, applies to three equally spaced ordinates y0, y1, y2 separated by a common interval h. The integral of the function over the interval [x0, x2] is approximated as:

Area ≈ (h ÷ 3) × (y0 + 4y1 + y2)

The rule is exact for polynomials up to degree three, which is sufficient for the smooth curves typical of ship sections and waterplanes. Extended to n + 1 ordinates where n is even, the multiplier sequence becomes 1, 4, 2, 4, 2, …, 2, 4, 1 with a leading factor of h ÷ 3. For a ship divided into 10 equal intervals (11 stations), the multiplier sequence is 1, 4, 2, 4, 2, 4, 2, 4, 2, 4, 1. The Simpson’s first rule area integrator calculator and the displacement volume from Simpson’s first rule calculator both implement this scheme. The waterplane area from Simpson’s first rule calculator applies it specifically to waterplane integration.

Simpson’s second rule

Simpson’s second rule, the 1-3-3-1 rule or three-eighth rule, applies to four equally spaced ordinates. The integral approximation is:

Area ≈ (3h ÷ 8) × (y0 + 3y1 + 3y2 + y3)

The rule is also exact for polynomials up to degree three but is more convenient when an interval is subdivided into three equal parts - for example when a midship bay must be treated separately using a non-standard station spacing. Extended forms combine the first and second rules to handle interval counts that are multiples of three. The Simpson’s second rule integrator calculator provides numerical evaluation.

Trapezoidal rule

The trapezoidal rule approximates the area between two adjacent ordinates y0 and y1 separated by interval h as h × (y0 + y1) ÷ 2. Summed across all intervals, the area becomes h × (y0 ÷ 2 + y1 + y2 + … + yn−1 + yn ÷ 2). The trapezoidal rule is exact only for linear functions, so it introduces truncation error for curved hull forms and is less accurate than the Simpson rules for the same number of ordinates. In practice it is used when irregular station spacing prevents the application of Simpson’s rule, when checking calculation, or when a coarse first estimate is needed.

Application to waterplane area, sectional area, and volume

For the waterplane area at a given draft: the half-breadths at each station along the waterline are the ordinates, the station spacing is the common interval, and Simpson’s first rule gives the area of one side. The total area is twice the result.

For the immersed cross-sectional area at a given station: the half-breadths at each waterline level are the ordinates, the waterline spacing is the interval, and integration gives the area of the transverse section up to any waterline. This is precisely the quantity plotted in the Bonjean curves.

For displaced volume: the station cross-sectional areas at a given waterline are the ordinates, the station spacing is the interval, and integration gives the total submerged volume.

Longitudinal moments are computed by multiplying each ordinate by its distance from the reference point (usually the aft perpendicular) and integrating as before. Dividing the moment by the volume gives LCB.

Second moments and moments of inertia

The second moment (moment of inertia) of the waterplane area about the centreline is needed for BMT. Using Simpson’s first rule, each term in the sum is weighted by the cube of the half-breadth (since the second moment of a thin transverse strip of unit length about the centreline is b³ ÷ 12, where b = 2 × half-breadth, giving 2 × (b/2)³ ÷ 3 = b³ ÷ 12 per unit length). The ordinate for the second moment integral at each station is therefore (2/3) × half-breadth³, and the overall multiplier structure of Simpson’s rule is applied to these modified ordinates in the same way as for area.

The second moment about the transverse axis through the centre of flotation, needed for BML, requires a parallel axis shift. The second moment about the aft perpendicular is computed first by including a lever-squared term (x²) in each ordinate contribution. The parallel axis theorem then gives the second moment about the centroid (at LCF) as the second moment about the aft perpendicular minus Aw × LCF².

Gauss’s divergence theorem and modern volume integration

In modern mesh-based hydrostatics programs, the displaced volume and its moments are computed by applying Gauss’s divergence theorem: the volume integral of a divergence-free vector field over the submerged volume is converted to a surface integral over the bounding surface. For a flat-bottomed waterplane, the volume below a horizontal waterline can be expressed as a surface integral over the hull panels below that waterline plus the waterplane. This allows the exact submerged geometry to be evaluated at arbitrary drafts without discretising into transverse stations, and it forms the mathematical basis of all modern panel-method hydrostatics codes.

Bonjean curves

Definition and construction

Bonjean curves (named for the nineteenth-century French naval constructor Antoine-Charles Bonjean, who published the technique in 1844) are a set of curves, one per transverse station, each showing how the immersed cross-sectional area at that station varies with draft. The horizontal axis represents cross-sectional area in square metres; the vertical axis represents draft in metres measured above the keel; and the curve itself is the running integral of the half-breadth-times-two at each waterline level up to the stated draft. Equivalently, the slope of the Bonjean curve at any draft equals the local waterline breadth at that station and draft.

The construction procedure for each station is: for a series of waterlines from the keel to the deck, calculate the immersed half-breadth at that station and waterline from the body plan offsets, then numerically integrate (using Simpson’s rule or the trapezoidal rule) from the keel upward to accumulate the total cross-sectional area at each waterline height. The resulting series of (draft, area) pairs is plotted or tabulated as the Bonjean curve for that station.

Reading a trimmed waterline

The key capability of the Bonjean curve set is that displacement at an arbitrary, non-parallel waterline can be read off directly without recomputing the integration. Consider a vessel trimmed by the stern: the aft draft exceeds the forward draft. For each station, the actual draft at that station is determined from the trim geometry - the waterline intersects the station at a draft that varies linearly (or nearly so, for small trim) from forward to aft. The cross-sectional area at each station is then read directly from the Bonjean curve for that station at the appropriate draft. The station areas are then integrated along the length using Simpson’s rule to give total displaced volume. The first moment of station areas, divided by total volume, gives LCB. This procedure is exact regardless of the magnitude of trim, making Bonjean curves essential for vessels with significant trim - tankers leaving port, bulk carriers loading, naval vessels with unusual load distributions.

For a damaged ship with flooding restricted to specific compartments, the waterline in the damaged condition is typically trimmed and possibly inclined simultaneously. The Bonjean curves, applied to the intact portions of the hull, give the buoyancy distribution along the hull, from which the reserve buoyancy and the trim in the damaged condition can be found. This method predates computer programs and remains the basis on which early damage stability analysis was carried out manually.

Application to longitudinal strength

Before the vessel departs on each voyage, the hull-girder bending moment acting along its length must be estimated to confirm it falls within the permissible values set by the classification society. The still-water bending moment (SWBM) at any section is the net effect of the distributed weight of the ship (cargo, structure, machinery, fuel, ballast) and the distributed buoyancy force. Bonjean curves provide the buoyancy distribution: at each station, the buoyancy force per unit length is ρ × g × (Astation), where Astation is the immersed cross-sectional area read from the Bonjean curve at the local draft. The difference between the cumulative buoyancy distribution and the cumulative weight distribution gives the shear force, and integrating the shear force gives the bending moment. IMO, IACS, and classification societies prescribe permissible SWBM values (IACS Unified Requirement S11, still-water bending moment calculator) and required section moduli (required section modulus calculator) to ensure hull-girder safety.

Hogging occurs when the midship region is more buoyant per unit length than the ends - typical of a lightly loaded vessel floating on a wave crest amidships. Sagging occurs when the ends carry more buoyancy than the midship - typical of a fully laden vessel or one on a wave trough amidships. The required section modulus for the hull girder is sized to withstand both the permissible still-water bending moment and the wave bending moment (wave bending moment calculator).

Beach landing condition

Amphibious vessels, including landing ship tanks (LSTs), hovercraft support craft, and some offshore supply vessels, must ground intentionally during cargo operations. The beach landing condition places the bow resting on a gradient seabed with part of the hull unsupported. The Bonjean curves give the buoyancy available at each section at the actual local draft, and the distribution of buoyancy minus weight gives the reaction force at the keel blocks or the touchdown point. The beach landing intact stability margin calculator addresses the stability criterion in this condition.

Bonjean curves in damage stability analysis

After a collision or grounding, the surviving hull must support the ship against sinking. The damaged condition is characterised by flooded compartments whose buoyancy contribution has been lost. To find the equilibrium waterline after damage, the naval architect or stability officer must find the waterline (described by its mean draft, trim, and heel) at which the buoyancy of the intact, unflooded portions of the hull exactly supports the total ship mass, with the centre of buoyancy directly below the centre of gravity.

The Bonjean curves allow this search to be conducted iteratively. For a trial waterline defined by a mean draft, trim, and heel, the draft at each station is computed from the geometry of the inclined waterplane. At each station, the immersed cross-sectional area of the intact (non-flooded) portion is read from the Bonjean curve at the local draft. These areas are integrated to give the total intact buoyancy volume. If the buoyancy force ρ × g × *∇*intact equals the ship’s displacement, and if the centres of buoyancy and gravity are aligned vertically, the trial waterline is the equilibrium position. If not, the trial waterline is adjusted and the process repeated.

This Bonjean-curve iteration is the manual precursor to the lost-buoyancy method implemented in modern stability programs. The damage stability article describes the regulatory framework; the residual GM check calculator supports the stability assessment after flooding.

Free trim versus even-keel hydrostatics

Even-keel hydrostatics assume the vessel floats with zero trim: the tabulated draft is the same at all stations. This is the standard presentation in most hydrostatic tables and is the format used during draft surveys when corrections for trim are applied afterward. Free-trim hydrostatics allow the vessel to adopt whatever trim brings the centre of buoyancy directly below the centre of gravity, and the resulting table gives Δ, LCB, and LCF at the natural trimmed condition. Modern software generates both; the even-keel table remains the commercial standard but Bonjean-based calculations implicitly give free-trim results.

Draft survey application

The commercial draft survey is the primary means by which cargo surveyors at loading and discharging ports determine how much cargo was loaded or discharged. The surveyor reads the six or eight draft marks (forward and aft on each side, sometimes also amidships) and computes the mean draft at the forward and aft perpendiculars, correcting for the position of the draft marks relative to the perpendiculars. The density of dock water is measured by hydrometer.

The mean draft is entered into the hydrostatic table to read displacement, using the even-keel table with a trim correction. The trim correction for displacement uses the formula: correction = trim × LCF offset from amidships ÷ Lpp, multiplied by the first moment of displacement with trim, which approximates to (trim × Tpc × LCF distance from midship) expressed as a change in displacement. Accurate application requires the tabulated value of LCF at the observed draft, which is why the hydrostatic table must be on board and available.

The trim from weight shift calculator provides the inverse: given a known weight shift, it computes the resulting trim change.

Cross curves of stability and GZ

From Bonjean to righting arm

The GZ righting lever is the perpendicular distance between the line of action of buoyancy and the line of action of gravity when the vessel is heeled. At small angles of heel (typically below 10–12°) this is closely approximated by GMT × sin θ, but at larger angles the relationship is non-linear and hull form effects dominate. The full GZ curve must be computed from first principles by computing displaced volume and the position of the centre of buoyancy at each heel angle.

KN cross curves

The cross curves of stability - universally known as KN curves - are a compact way to pre-compute the righting moment information for all displacements and all relevant heel angles. For each heel angle θ in the set {10°, 20°, 30°, 40°, 50°, 60°, 70°, 80°} the shipyard’s hydrostatics software computes, for every value of displacement Δ across the full range, the quantity KN(θ, Δ), which is the righting lever measured from the keel K to the projection N of the centre of buoyancy onto the heeled vessel’s centreline plane. Plotting KN against Δ for each θ gives the cross curve family. The GZ at any condition is then recovered as:

GZ = KN − KG × sin θ

where KG is the vessel’s actual centre of gravity height above the keel in the loading condition being assessed. The GZ from KN cross-curve calculator performs this step for a specified KG and displacement.

The GZ curve against heel angle is then used to evaluate the intact stability criteria of the IS Code: minimum area under the GZ curve to 30°, minimum area to 40° or to the flooding angle, minimum GZ at 30°, angle of maximum GZ, and initial GMT. Any loading condition that fails a criterion requires the operator to move ballast or reduce cargo.

Dynamical stability

The area under the GZ curve from 0° to any angle θ is the dynamical stability - the energy per tonne of displacement required to heel the vessel to that angle. It is directly relevant to the ship’s ability to survive a sudden gust of wind or a wave-induced roll impulse. The dynamical stability area under GZ calculator integrates the GZ curve numerically and outputs the stability energy at any target angle.

Weather criterion

The IS Code general criterion C requires that the area under the GZ curve between the wind heeling arm and the restoring GZ curve satisfies a reserve ratio known as the weather criterion. The wind heeling arm is a parabolic function of heel derived from the steady wind pressure assumed to act on the vessel’s lateral windage area above the waterline. The steady heel angle produced by this wind lever is found by the intersection of the GZ curve and the wind lever; the vessel must then have sufficient reserve stability beyond that angle to withstand a rolling motion assumed to take the vessel to a windward angle of at least 25° or the angle of the first weather deck immersion, whichever is smaller.

The weather criterion is evaluated using the same cross curves of stability and GZ computation described above, but requires additional input of the windage area and its centroid above the waterline. These quantities are determined from the general arrangement plan and vary significantly with deck cargo (containers, timber, vehicles) and with the amount of superstructure. Container ships and ro-ro ferries are typically more sensitive to the weather criterion than bulk carriers or tankers of similar displacement.

Wall-sided formula

For heels up to the angle at which the deck edge enters the water, the wall-sided approximation gives a convenient closed-form expression for GZ. The wall-sided formula states:

GZ = sin θ × (GMT + BMT × tan²θ ÷ 2) − KG × sin θ

simplified as GZ = sin θ × (GMT + (1/2) × BMT × tan²θ) when GMT is used net of KG. This expression shows that even when initial GMT is slightly negative (a condition of initial loll), a vessel can still have positive GZ at larger angles due to the tan²θ term provided BMT is sufficiently large. The angle of loll is then the angle at which the wall-sided GZ first becomes positive again. The wall-sided formula is most accurate for full-form vessels such as tankers and bulk carriers and becomes unreliable as deck immersion approaches. The wall-sided heel calculator computes the GZ and angle of loll directly from GMT and BMT.

Inclining experiment

Purpose and procedure

The inclining experiment is the direct measurement of a vessel’s GMT and, by subtraction of the computed BMT from the hydrostatic table, the height of the centre of gravity KG. It is performed on the completed (or substantially completed) vessel, typically in a shipyard or dry dock, when the mass and position of every item of variable load is known precisely or can be removed or accounted for.

The procedure involves shifting a known test mass w (usually lead ingots or steel plates) through a measured transverse distance d across the deck. The resulting heel angle θ is measured by long pendulums or inclinometers. The formula GMT = (w × d) ÷ (Δ × tan θ) gives the metacentric height. Dividing w × d by Δ × tan θ must be repeated for at least four mass positions to give a reliable average and to detect asymmetries.

From GMT the lightship KG is derived as KG = KB + BMT − GMT, with KB and BMT taken from the hydrostatic table at the observed inclining draft. The lightship displacement ΔL is read from the hydrostatic table at the observed draft, corrected for density and trim. The longitudinal centre of gravity LCG is found from the trim at the time of the inclining experiment through LCG = LCB + trim × MCT1cm ÷ Δ. The GM from inclining experiment calculator and the inclining experiment GM from pendulum calculator both support this analysis.

Measurement of heel angle

Two instruments are used to measure the heel angle in an inclining experiment. Long pendulums - typically three or four, positioned at widely separated locations on the vessel - measure the tangent of the heel angle as the ratio of the horizontal excursion of the pendulum bob to the pendulum length. Pendulums of 3–5 m length give high sensitivity. Inclinometers (spirit levels or electronic tiltmeters) provide a check and are now often used as the primary instrument on smaller vessels. Both instruments must be calibrated and their readings corrected for any initial list. IMO guidelines (MSC.267(85), Appendix 2) prescribe minimum requirements for the number of measurement points, the range of mass shifts, and the treatment of draft readings and density measurements.

Conditions and corrections

For the inclining experiment to give a valid lightship KG, the vessel must be as close to completion as possible. All items that will be in the lightship must be on board in their permanent positions; all items that will be removed before delivery (staging, temporary equipment, calibration loads) must have been removed or accurately weighed and measured. Liquids in tanks must be either pressed up (full, with no free surface) or empty; any liquid with a free surface introduces a free surface correction that must be calculated and applied. The total free surface correction iT × ρL ÷ Δ × ρ is subtracted from the measured GMT to give the effective GMT.

Personnel not involved in the test must leave the vessel to avoid uncontrolled weight movements. Any wind must be light enough that the vessel is not pushed to a stable heel by wind pressure. The experiment is typically conducted at the calm water of an inner harbour or dry dock with the vessel moored freely.

Inclining survey report

The approved inclining report documents the vessel’s lightship displacement, KG, LCG, and the number of items in the ship’s inventory list that were on board during the experiment. Classification societies and flag state administrations retain copies of the approved report. When the vessel undergoes a later weight survey (lightweight check) - required by some class rules and flag states at intervals of five years for passenger ships - the survey result is compared against the approved inclining report to detect whether accumulation of structural additions, coatings, or trapped water has shifted KG beyond the approved value.

Regulatory requirement

The IS Code and SOLAS II-1 require that an inclining experiment be carried out at the completion of every new ship and after any alteration significant enough to affect stability. Passenger ships must be re-inclined after major alterations. The lightship particulars derived from the inclining experiment are the foundation of the approved stability booklet: every other loading condition in the booklet is derived from the lightship KG and LCG by adding items of variable load.

Classification societies such as Lloyd’s Register, Bureau Veritas, and the American Bureau of Shipping witness the inclining experiment and approve the results. IACS Unified Requirement S1 specifies the longitudinal strength calculations, which depend on the lightship LCG.

Loading manual and loading instrument

Statutory basis

Every ship subject to SOLAS chapter II-1 regulation 5-1 (cargo ships above a specified size) must carry a loading manual approved by the flag Administration or an authorised classification society. The loading manual contains the hydrostatic data, the permissible loading conditions, the KG limit curves, and - for bulk carriers, ore carriers, and combination carriers - the structural envelope within which loading and ballasting sequences must be kept.

IACS UR S1 requires that the design still-water bending moment and shear force envelopes be determined during the design phase and that the approved loading manual include sufficient conditions to demonstrate compliance. UR S11 provides the formulae for the permissible SWBM and wave bending moment as functions of ship length and type.

For vessels exceeding 150 m in length, a loading instrument (a dedicated computer or electronic calculator) is often required or accepted in lieu of pre-calculated conditions. The instrument must compute the displacement, trim, stress, and stability for any loading condition the officer enters, checking automatically against all permissible limits.

KG limit curve

The KG limit curve plots, for each displacement Δ, the maximum KG that satisfies all intact stability criteria (including the IS Code general criteria and any ship-type-specific criteria such as the weather criterion). Any loading condition whose (Δ, KG) point lies below the KG limit curve is acceptable; any condition above the curve is prohibited. The KG limit intact calculator computes this boundary for a user-supplied stability criterion.

New KG calculations

During cargo operations and voyage planning, the ship’s officer must track how each addition, removal, or transverse shift of mass affects KG and LCG. The relevant relations are: new KG = (old Δ × old KG + added mass × height of added mass) ÷ new Δ for addition, and symmetric for removal. The new KG after weight addition calculator performs this step. The metacentric shift calculator computes the change in GM from a vertical weight shift.

Practical hydrostatic properties - worked relationships

Bodily sinkage and rise

When a mass m tonnes is added to a vessel floating in water of density ρ, the vessel sinks bodily by s = m ÷ Tpc centimetres at the centre of flotation, plus a trim correction. The total change in draft at any point is the bodily sinkage plus or minus the change in trim multiplied by the distance from that point to LCF. This separation of bodily sinkage from trim is the basis of the standard draft survey procedure used at loading ports worldwide.

For a loaded bulk carrier with Tpc = 85 t/cm, loading 5,000 t of additional ore causes a bodily sinkage of 5,000 ÷ 85 = 58.8 cm ≈ 59 cm before any trim adjustment. If the cargo is loaded 10 m aft of LCF, the trim change is 5,000 × 10 ÷ MCT1cm cm. With MCT1cm = 520 t⋅m/cm, the trim change is 50,000 ÷ 520 = 96 mm ≈ 10 cm by the stern, resulting in an aft draft increase of approximately 59 + (fraction forward of aft perpendicular × 10) cm and a forward draft change of 59 minus the remaining trim fraction.

Density correction

When a vessel moves from salt water (1.025 t/m³) to fresh water (1.000 t/m³), the displacement Δ stays the same but the volume ∇ must increase by the factor 1.025 ÷ 1.000, so the vessel sinks by FWA. The precise FWA is computed at the summer load draft and is marked on the load line. The dock water allowance for any intermediate density ρd is DWA = FWA × (1.025 − ρd) ÷ (1.025 − 1.000).

TPC at different densities

The Tpc value in the hydrostatic table is conventionally stated for salt water. In fresh water, the lower density means each centimetre of immersion adds less mass to maintain the buoyancy, so Tpc(fw) = Tpc(sw) × 1.000 ÷ 1.025 ≈ 0.976 × Tpc(sw). For intermediate densities, direct proportion applies. The waterplane area coefficient calculator and the waterplane area Simpson’s first rule calculator provide the waterplane area from which Tpc is derived.

Interaction between trim and stability

Operating a vessel at significant trim affects the apparent GMT in two ways. First, the actual KB and BMT at the trimmed condition differ slightly from the even-keel table values at the same displacement, because the waterplane and submerged volume geometry change with trim. For small trims (less than 0.5% of Lpp) this effect is negligible. For larger trims, either the free-trim hydrostatics table must be used or a trim correction to GMT must be applied using the rate of change of the waterplane second moment with trim.

Second, the loading officer must verify that the trim is within the approved range given in the loading manual. Excessive trim by the stern can cause propeller emergence and racing on fast vessels, and in extreme cases can place the stern anchor or stern accommodation at risk of wave damage. Excessive trim by the head can impair forward visibility below the requirements of SOLAS chapter V regulation 22. The trim and list article covers the operational aspects; the trim from loading centroid calculator computes the equilibrium trim for any cargo distribution.

Rolling period and GM estimation

When the hydrostatic table is not available or when a quick check of GMT is needed, the rolling period method provides an approximate estimate. The natural rolling period T of the ship in seconds is related to GMT by T = 2 × Cr × B ÷ √GMT, where B is the ship’s breadth and Cr is an empirical coefficient typically in the range 0.78–0.82 for loaded cargo ships, 0.73–0.76 for loaded tankers, and 0.85–0.90 for lightly loaded vessels. The rolling period GM calculator inverts this formula to estimate GMT from an observed rolling period. This method is used as a sanity check and also as a rough measure after damage, when the precise loading condition may not be known.

Damage condition hydrostatics

Loss of waterplane area

When a compartment is flooded, the effective waterplane area decreases by the area of the flooded compartment multiplied by the permeability of that compartment. Permeability accounts for the proportion of the compartment volume that is actually available for flooding - solid cargo has a permeability near 0.60, machinery spaces near 0.85, and empty tanks near 0.95–0.97. The reduced waterplane area lowers Tpc, raises the draft, and shifts LCF, altering all subsequent trim and stability calculations. The reduction in Tpc means each tonne added or removed causes a larger draft change than in the intact condition, which is relevant to emergency de-ballasting or counter-flooding operations.

Residual GM and residual stability

The damage stability requirements of SOLAS chapter II-1 and the relevant parts of the IS Code specify minimum residual GM and minimum GZ criteria in the damage condition. These are evaluated using the damaged waterline hydrostatics, which are obtained from the Bonjean curves by removing the contribution of flooded compartments and re-integrating. The residual GM check calculator is a direct application of this method. Cross-link to intact stability for the undamaged stability framework.

For passenger ships, SOLAS chapter II-1 part B-2 regulation 7 (for ships built before 1 January 2009) and the harmonised damage stability regulations (regulation 7-2 onward) prescribe specific residual GZ requirements varying with ship type, number of passengers, and extent of assumed damage. Oil tankers are additionally governed by MARPOL Annex I regulations 27 and 28, which require two-compartment flooding survivability for new vessels above a threshold length.

Lost buoyancy versus added weight methods

Two mathematical methods are used in damage stability analysis. The lost-buoyancy method treats the flooded space as if it had been removed from the hull - the displacement is unchanged, but the hull now has a smaller effective volume. The waterplane area, Tpc, MCT1cm, and the centre of flotation are all recomputed from the reduced effective waterplane. The added-weight method treats the flood water as an additional mass added to the ship, with an associated free surface correction. Both methods give the same equilibrium waterline when applied correctly, but the lost-buoyancy method is more suited to Bonjean-curve analysis, while the added-weight method is more intuitive for officers using a loading instrument.

Free surface effect

Liquid in a partially filled tank reduces effective GM by adding a virtual rise of G of value ρL × iT ÷ (ρS × Δ), where ρL is the density of the liquid, iT is the second moment of the free surface area about its centreline, ρS is the density of the salt water in which the ship floats, and Δ is the displacement. This free surface effect is additive over all slack tanks and must be included in every stability calculation in the approved loading instrument.

Ship-type hydrostatic characteristics

Bulk carriers and ore carriers

Bulk carriers present a distinctive hydrostatic profile. A Capesize bulk carrier of approximately 180,000 DWT has a very full form (Cb approximately 0.83–0.85) and a deep, wide midship section with high Cm (typically 0.98–0.99). The KM curve typically peaks at a draft near 6–8 m and then falls steeply as the full-form hull does not gain waterplane area fast enough to offset the growth of ∇. At the maximum design draft (around 18 m), KM may be only 0.5–1.0 m above the keel, meaning that for the vessel to satisfy the minimum GM criterion of 0.15 m (IS Code), the actual KG must be no higher than about KM − 0.15 m. Heavy ores loaded into just one pair of holds concentrate weight at a low position and give a generous GM; light grain or coal loaded into all holds presses KG upward and can bring the vessel close to the limiting KM curve. This is why alternating hold loading - odd-numbered holds only - is often the critical structural loading case for a bulk carrier while the stability is comfortable, whereas full hold loading may be critical for stability.

Oil tankers and product tankers

Oil tankers of VLCC and ULCC size have very full forms (Cb 0.83–0.88) and extremely high Tpc values. A 320,000 DWT VLCC has a Tpc at the load waterline in the range 200–220 t/cm. The waterplane area is correspondingly enormous - approximately 20,000 m² at load draft. These vessels have generous initial stability (GMT typically 3–7 m fully loaded) because the very large BMT from the wide beam dominates KG changes. The primary stability challenge occurs in the ballast condition, when the vessel floats at perhaps half its loaded draft with empty cargo tanks. In this condition ∇ is small relative to IT, so BMT is large, but KB is low. The ballast condition for a VLCC typically gives KMT in excess of 10 m and KG in the range 8–9 m, giving GMT of 1–3 m - acceptable but requiring careful ballast tank management to avoid shifting KG upward.

Free surface effect from ballast tanks is the primary stability concern for tankers in ballast. With double-bottom ballast tanks filling to only 50% capacity, the free surface correction from each tank can be several centimetres of GM reduction. High-permeability tanks (large plan area) in the double bottom are particularly significant. The free surface effect must be computed for each tank arrangement.

Container ships

Container ships have moderate Cb values (0.60–0.73) and a distinctive deck arrangement that allows containers to be stacked five to six high above the hatch coaming. The vertical centre of gravity KG is therefore much higher relative to the ship’s dimensions than for a bulk carrier or tanker. Container vessels also carry their stability certification based on a maximum KG curve (or equivalently, a minimum GMT curve) that accounts for both the IS Code general criteria and the weather criterion. At low displacement (partial loads) the vessel may have low GMT because the waterplane area is small, and at full displacement the stacked containers raise KG. The KG limit curve is therefore non-linear, with a minimum at intermediate displacements. Port state control has detained container ships for inadequate stability arising from undeclared deck cargo or higher-than-declared container weights - an issue addressed by the SOLAS chapter VI regulation 2 container VGM (verified gross mass) requirements.

Passenger ships

Passenger ships require the most elaborate stability calculations because of the width of the loading range (from full passenger load with all stores, to passenger embarkation or disembarkation in port, to various damage scenarios). They are subject to both the IS Code general criteria and ship-type-specific criteria in the IS Code part B, including special provisions for the effect of passengers crowding to one side (heeling moment from passenger crowding) and for the effect of fire-fighting water accumulation on open deck areas. The inclining experiment for a large cruise ship is a major undertaking, often involving test masses of 500–1,000 t and pendulums 5 m in length, with the class surveyor and flag state representative in attendance.

Modern computational hydrostatics

Station-based numerical methods

In the pre-computer era, hydrostatic tables were computed by a draughtsman measuring offsets from the lines plan with a rolling planimeter or Amsler integrator, then applying Simpson’s rule by hand. A full hydrostatic table for a new vessel took several weeks to complete. Computing machines from the 1960s onward automated the arithmetic while retaining the station-and-waterline grid framework. The station-based method remains in wide use in regulatory calculations and is the basis of most class-approved loading instruments, because it is transparent, auditable, and understood by surveyors worldwide.

Mesh-based volume integration

Modern surface-modelling hydrostatics programs represent the hull as a triangulated or parametric surface mesh rather than a grid of stations and waterlines. The displaced volume is then computed by integrating over all mesh panels that fall below a specified waterplane. This approach handles re-entrant forms, bulbous bows, tunnel sterns, and other complex geometries that defeat the simple station-integration approach. Accuracy depends on mesh density: typical commercial programs use adaptive refinement near the waterline.

NURBS and IGES geometry

Non-uniform rational B-spline (NURBS) surfaces, imported via IGES or STEP files, are the current standard for hull geometry transfer between design programs. A NURBS hull can be interrogated analytically for waterplane intersection without discretisation, giving exact (within floating-point precision) sectional areas and moments. Commercial packages interrogate the NURBS geometry to derive Bonjean curve data and cross curves of stability at arbitrarily fine intervals.

Panel methods and resistance coupling

Panel method codes discretise the hull surface into flat quadrilateral panels and solve the potential-flow boundary-value problem to compute wave-induced pressure distributions. The hydrostatic solution (zeroth-order in wave amplitude) is a by-product, giving the still-water displacement and pressure distribution. More advanced frequency-domain strip theory and three-dimensional diffraction codes require the station-based hydrostatic data as input, creating a direct bridge between the hydrostatic table and seakeeping and structural analysis.

Commercial software

The principal commercial packages for ship hydrostatics are Maxsurf Stability (Bentley), NAPA (NAPA Ltd, Finland), AVEVA Marine (AVEVA Group), and Rhino 3D with the Orca3D plugin. Free and open-source options include FreeShip (now largely superseded) and DELFTship (DELFTship.net), which support Bonjean curves, cross curves, and full IS Code criteria evaluation. Class-approved loading instruments typically run proprietary hydrostatic kernels validated against the approved ship-specific data. The ShipCalculators.com calculator catalogue provides web-based access to individual hydrostatic sub-calculations without requiring installation of full naval architecture software.

Relationship to regulatory frameworks

Load line

The load line is defined, at its most fundamental level, by the freeboard required to maintain adequate reserve buoyancy and a minimum standard of stability. Freeboard is the distance from the summer load waterline to the freeboard deck at side. The hydrostatic data - specifically Tpc, Cb, and displacement at the load draft - directly determine the tabular freeboard corrections applied by the classification society when assigning the vessel’s load line marks. The reserve buoyancy calculator computes the freeboard-based reserve buoyancy percentage.

Intact stability criteria

The IMO Intact Stability (IS) Code 2008 (MSC.267(85)) prescribes minimum values for GMT, GZ at 30°, the area under GZ, and the angle of maximum GZ for a range of ship types including general cargo ships, bulk carriers, container ships, and offshore vessels. These criteria are evaluated from the hydrostatic table and the cross curves of stability for each loading condition. The metacentric height GM calculator computes GMT from the hydrostatic inputs for any draft and KG.

MARPOL and class requirements

MARPOL Annex I regulation 27 includes intact stability criteria for oil tankers, cross-referencing the IS Code. Regulation 28 covers subdivision and damage stability for oil tankers. Both regulations are enforced through Port State Control inspections, which verify that the vessel’s approved stability booklet and loading instrument are on board and being used. Related MARPOL convention requirements for bulk carriers and combination carriers similarly mandate longitudinal strength monitoring tied directly to the hydrostatic data.

IACS longitudinal strength

IACS UR S1 (requirements for longitudinal strength) and UR S11 (longitudinal hull-girder loads) require the shipyard to compute and document still-water bending moments and shear forces for a representative set of loading conditions. The permissible SWBM limits are expressed as fractions of the wave bending moment computed from the S11 formula. All these calculations originate in the hull’s hydrostatic data and the Bonjean curves. Classification society approval of the loading manual constitutes verification that the stated loading conditions satisfy both the stability and the structural envelope simultaneously.

Classification and survey

Classification societies, including Lloyd’s Register, Bureau Veritas, DNV, American Bureau of Shipping, ClassNK, and Korean Register, require the hydrostatic data to be produced and verified during the plan approval stage of a new build. Surveyors attending the inclining experiment witness the measurement and check the consistency of the observed GM with the computed BM from the hydrostatic table. At every annual survey and special survey, the loading instrument is tested against pre-computed benchmark conditions to verify the hydrostatic kernel remains accurate. The classification society article describes the broader framework of survey and certification.

References

Archimedes, On Floating Bodies, ca. 250 BCE. Translation by T. L. Heath, The Works of Archimedes, Cambridge University Press, 1897.
Bonjean, A.-C., “Mémoire sur la construction des courbes qui représentent les sections transversales des navires à différents tirants d’eau”, Annales Maritimes et Coloniales, 1844.
Barras, C. B., Ship Stability for Masters and Mates, 6th edition, Butterworth-Heinemann, 2006.
Biran, A. B., Ship Hydrostatics and Stability, Butterworth-Heinemann, 2003.
IMO, International Code on Intact Stability 2008 (IS Code), Resolution MSC.267(85), International Maritime Organization, London, 2009.
IACS, Unified Requirement S1, Longitudinal Strength of Seagoing Ships, International Association of Classification Societies, 2015.
IACS, Unified Requirement S11, Longitudinal Strength Standard, International Association of Classification Societies, 2001 (rev. 2017).
IMO, SOLAS Consolidated Edition, Chapter II-1, Construction - Structure, Subdivision and Stability, Machinery and Electrical Installations, International Maritime Organization, London, 2020.
Molland, A. F. (ed.), The Maritime Engineering Reference Book, Butterworth-Heinemann, 2008.
Comstock, J. P. (ed.), Principles of Naval Architecture, Society of Naval Architects and Marine Engineers, New York, 1967.