Determine the longitudinal position of the buoyancy centre using hydrostatic moments or sectional areas with Simpson’s rule.
LCB from AP: — m
LCB as % of LPP from AP: — %
LCB from AP: — m
LCB as % of LPP from AP: — %
The longitudinal centre of buoyancy is the fore-and-aft position of the centroid of a ship's displaced underwater volume. It is the longitudinal point through which the resultant buoyant force acts in an upright hydrostatic condition.
LCB is a position rather than a volume or force. A reported value has no clear meaning unless its longitudinal datum, positive direction and loading condition are stated.
This calculator uses the aft perpendicular as its origin. Distances are measured forward from AP, so x = 0 at AP and x = LPP at FP.
The coordinate of the buoyancy centroid is the first longitudinal moment of displaced volume divided by the total displaced volume:
xB = ∫x dV ÷ ∇
On this page, xB is reported as LCB measured forward from AP.
| Symbol | Meaning | Usual unit |
|---|---|---|
| xB | Longitudinal coordinate of the centre of buoyancy | m |
| x | Longitudinal coordinate measured from the selected datum | m |
| dV | Element of displaced underwater volume | m3 |
| ∫x dV | First longitudinal moment of displaced volume about the datum | m4 |
| ∇ | Total displaced underwater volume | m3 |
| LPP | Length between aft and forward perpendiculars | m |
Dividing a first moment in m4 by a volume in m3 produces a longitudinal distance in metres.
Datum requirement: the first moment and the reported LCB must use the same origin and positive direction. This calculator assumes that the first moment is taken about AP with positive x measured forward.
Consider the following hydrostatic data:
LCBAP = 565,333.333 ÷ 8,800 = 64.242 m
LCB = 64.242 ÷ 120 × 100 = 53.535% of LPP from AP
The centre of buoyancy is therefore 64.242 m forward of AP.
| Method | Main inputs | Suitable use |
|---|---|---|
| Hydrostatic first moment | ∇, ∫x dV and LPP | Known hydrostatic volume and first-moment data |
| Station integration | LPP and equally spaced sectional areas | Offsets, sectional-area curves or preliminary hull geometry |
Use this method when displaced volume and its first longitudinal moment are available from hydrostatic calculations, a hull model or a table of form properties.
Both quantities must refer to the same draft, trim, hull geometry and longitudinal datum. A moment calculated about FP or amidships cannot be entered as though it were taken about AP.
The entered LPP does not change the LCB distance calculated from moment divided by volume. It is used to express that distance as a percentage of the AP–FP length and to check that the result lies within the stated perpendiculars.
The underwater volume can be represented as the integral of transverse sectional area along the ship:
∇ = ∫ A(x) dx
Its first longitudinal moment about AP is:
M1 = ∫ xA(x) dx
LCB is therefore:
LCB = ∫xA(x) dx ÷ ∫A(x) dx
For N equally spaced stations over LPP, the station spacing is:
Δx = LPP ÷ (N − 1)
The calculator applies Simpson's one-third rule to the sectional areas:
∇ ≈ Δx ÷ 3 × [A0 + An + 4(A1 + A3 + ... + An−1) + 2(A2 + A4 + ... + An−2)]
The same Simpson multipliers are then applied to the products xiAi to obtain the first moment.
Station requirement: Simpson's one-third rule requires an even number of intervals and therefore an odd number of equally spaced stations. Areas must be entered in order from AP to FP.
Consider seven equally spaced stations over:
Δx = 120 ÷ (7 − 1) = 20 m
| Station | x from AP (m) | A(x) (m2) | xA(x) (m3) | Simpson multiplier |
|---|---|---|---|---|
| 0 | 0 | 0 | 0 | 1 |
| 1 | 20 | 40 | 800 | 4 |
| 2 | 40 | 90 | 3,600 | 2 |
| 3 | 60 | 120 | 7,200 | 4 |
| 4 | 80 | 110 | 8,800 | 2 |
| 5 | 100 | 70 | 7,000 | 4 |
| 6 | 120 | 0 | 0 | 1 |
∇ = 20 ÷ 3 × [0 + 0 + 4(40 + 120 + 70) + 2(90 + 110)]
∇ = 8,800 m3
M1 = 20 ÷ 3 × [0 + 0 + 4(800 + 7,200 + 7,000) + 2(3,600 + 8,800)]
M1 = 565,333.333 m4
LCB = 565,333.333 ÷ 8,800 = 64.242 m from AP
LCB = 53.535% of LPP from AP
The same physical position can be reported in several ways. The datum and sign convention must accompany the number.
| Convention | Calculation | Reported value |
|---|---|---|
| Forward from AP | xB | 64.242 m from AP |
| Aft of FP | LPP − xB | 55.758 m aft of FP |
| Percentage from AP | xB/LPP × 100 | 53.535% LPP |
| Relative to amidships, positive forward | xB − LPP/2 | 4.242 m forward of amidships |
Some hydrostatic tables use amidships as zero and take aft as positive; others take forward as positive. Never infer the convention from the sign alone.
Do not compare bare numbers: an LCB reported as 4.2 m aft of amidships and one reported as 54% of LPP from AP may describe similar positions but use different origins and signs.
LCB describes the location of buoyancy, while the longitudinal centre of gravity, LCG, describes the location of the vessel's total weight.
If a loading change moves LCG away from the buoyancy line of action, an initial trimming moment is produced. The vessel changes trim, causing the underwater geometry and LCB to move until a new equilibrium condition is reached.
For this reason, simply subtracting LCB from LCG does not by itself provide the final change of trim. The calculation also requires displacement and a longitudinal hydrostatic stiffness quantity such as MCT 1 cm or longitudinal metacentric height.
Initial trimming moment = Δ × (LCG − LCB)
LCG and LCB must use the same datum and positive direction before this difference is calculated.
The longitudinal centre of buoyancy is the centroid of underwater volume. The longitudinal centre of flotation, LCF, is the centroid of waterplane area.
LCB is associated with the resultant buoyant force. LCF is the approximate pivot point for small changes of trim. They may lie close together on some hulls, but they are not interchangeable.
| Property | LCB | LCF |
|---|---|---|
| Geometric basis | Underwater volume | Waterplane area |
| Centroid formula | ∫x dV / ∇ | ∫x dAWP / AWP |
| Primary use | Buoyancy position and longitudinal equilibrium | Reference point for small trim changes |
LCB belongs to a particular immersed hull condition. As draft changes, the sectional-area curve changes and its centroid may move forward or aft.
Trim also changes the underwater shape. A sectional-area dataset taken from an even-keel condition should not automatically be used to represent a substantially trimmed condition.
Internal weight transfers do not directly alter total displacement, but they can move LCG and cause the vessel to trim. The resulting change in underwater geometry then changes LCB.
Simpson integration approximates the area curve between the supplied stations. Its accuracy depends on station spacing and how well the data represent rapid geometric changes.
Additional stations may be needed near:
The current calculator requires equal spacing over LPP. It does not support arbitrary station coordinates or mixed Simpson multipliers.
LCB is useful for hydrostatic reporting, preliminary hull-form studies, longitudinal balance checks and trim calculations.
LCB alone cannot determine:
Result check: when x is measured forward from AP over LPP, the calculated LCB should lie between 0 and LPP. A result outside that range normally indicates a datum, unit, station-order or moment error.
The definitions and numerical methods used on this page follow established naval-architecture sources:
NauticalSolver calculators are intended for preliminary engineering, study and independent checking. Use approved hydrostatic particulars, vessel-specific geometry and the loading computer for operational, contractual or statutory trim calculations.