Calculation
GM = IW/V − BG
Positive metacentric height indicates a restoring tendency at small heel angles. It is an initial-stability measure, not a complete capsize or approval check.
Anfangsstabilität schwimmender Körper · GM = IW/V − BG
Calculate a Floating Body's Metacentric Height
Positive metacentric height indicates a restoring tendency at small heel angles. It is an initial-stability measure, not a complete capsize or approval check.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Metacentric height GM — Vertical separation from centre of gravity G to metacentre M. Positive means initially restoring, zero neutral and negative initially overturning; further stability checks are needed for design.
- Waterplane second moment IW — Second moment of the waterplane area about the considered heel axis through its area centre. Compute from geometry or CAD; for a rectangle of length L and beam b rolling about its lengthwise axis, IW = Lb³/12.
- Displaced liquid volume V — Volume of the submerged body in the considered floating equilibrium. Obtain from draft and geometry or from mass and liquid density.
- Centre separation BG — Vertical distance from centre of buoyancy B, the centre of displaced volume, to centre of gravity G. Positive when G is above B; negative when G is below B.
Example
With IW = 2.667 m⁴, V = 4 m³ and BG = 0.20 m, GM = 2.667/4 − 0.20 = 0.467 m: small heel angles have a restoring tendency.
Assumptions and limits
Floating equilibrium with weight equal to buoyancy, small heel angles and first-order fixed waterplane geometry. Free liquid surfaces on board, large angles, changing cargo, dynamic waves and regulatory safety criteria are excluded.
Understand Metacentric height and initial stability
Will a floating body return after a small sideways disturbance? Metacentric height describes this initial tendency.
What does this quantity describe?
A floating body displaces liquid of equal weight. B is the centre of buoyancy, the centre of displaced volume; G is the body's centre of gravity. At a small heel angle, the shifted buoyancy line intersects the original vertical at the metacentre M. Metacentric height GM = IW/V − BG measures from G to M. IW is the waterplane second moment about the heel axis, V the displaced volume and BG the signed vertical distance from B to G.
Formula and variables
GM = IW/V − BG
Metacentric radius: BM = IW/VMetacentric height: GM = BM − BGInitial tendency: GM > 0 restoring; GM < 0 overturning
| Symbol / input | Meaning |
|---|---|
| Metacentric height GM | Vertical separation from centre of gravity G to metacentre M. Positive means initially restoring, zero neutral and negative initially overturning; further stability checks are needed for design. |
| Waterplane second moment IW | Second moment of the waterplane area about the considered heel axis through its area centre. Compute from geometry or CAD; for a rectangle of length L and beam b rolling about its lengthwise axis, IW = Lb³/12. |
| Displaced liquid volume V | Volume of the submerged body in the considered floating equilibrium. Obtain from draft and geometry or from mass and liquid density. |
| Centre separation BG | Vertical distance from centre of buoyancy B, the centre of displaced volume, to centre of gravity G. Positive when G is above B; negative when G is below B. |
Choose the inputs correctly
Calculate IW in m⁴ from the waterplane outline about the considered heel axis; for a rectangle of length L and beam b rolling about its lengthwise axis, IW = Lb³/12. V in m³ is the displaced volume at the actual floating equilibrium, found from draft and geometry or mass divided by fluid density. BG in metres is positive when G lies above B and negative when G lies below. Determine both centres from mass and submerged-volume distributions.
How to use the calculator
Use the same heel axis throughout. Enter waterplane second moment and displacement for the actual loading condition, then signed BG. Positive GM indicates a restoring tendency at small angles; operation and approval need further stability criteria.
Worked example
For IW = 2.667 m⁴, V = 4 m³ and G located BG = 0.20 m above B, IW/V = 0.667 m and GM = 0.667 − 0.20 = 0.467 m. Small heels restore in this model.
Understand the result and units
GM > 0 means initial stability, GM = 0 neutral balance and GM < 0 an initially overturning tendency. A positive result says nothing by itself about large angles, waves or shifting cargo.
IW uses m⁴, V m³, and BG and GM metres internally. The ratio IW/V has units of metres. Signs of BG and GM matter to interpretation.
Typical applications
Early plausibility checks for pontoons, floating platforms and floating tanks; comparing waterplane beam and centre-of-gravity position.
Assumptions, limits and common mistakes
Floating equilibrium and small heel angles only. Free liquid surfaces in tanks, large heel, changing cargo, strongly varying geometry, currents and waves are excluded. This is no complete capsize or maritime regulatory check.
Common mistake: IW must be about the considered heel axis; do not swap length and beam in Lb³/12. BG is signed: it is negative when G lies below B. Displaced volume is not the body's full envelope volume.
Frequently asked questions
What is “Metacentric height and initial stability” used for?
Early plausibility checks for pontoons, floating platforms and floating tanks; comparing waterplane beam and centre-of-gravity position.
Where do the input values come from?
Calculate IW in m⁴ from the waterplane outline about the considered heel axis; for a rectangle of length L and beam b rolling about its lengthwise axis, IW = Lb³/12. V in m³ is the displaced volume at the actual floating equilibrium, found from draft and geometry or mass divided by fluid density. BG in metres is positive when G lies above B and negative when G lies below. Determine both centres from mass and submerged-volume distributions.
What does the result not cover?
Floating equilibrium and small heel angles only. Free liquid surfaces in tanks, large heel, changing cargo, strongly varying geometry, currents and waves are excluded. This is no complete capsize or maritime regulatory check.