Buoyancy & Stability (GM) Calculator

Important: This tool provides an educational, first-pass estimate of initial transverse metacentric height (GM) for small heel angles. It does not replace a stability booklet, class rules, an inclining experiment, or professional naval architecture review. Vessel stability is safety-critical; use this calculator to build intuition and to sanity-check inputs, not to certify seaworthiness.

What this vessel GM calculator estimates

For a vessel at a slight heel, the shift in buoyancy can create either a restoring or overturning moment. Initial transverse metacentric height, or GM, summarizes that near-upright response:

• GM > 0 generally indicates positive initial stability (a restoring tendency for small heel angles).
• GM ≈ 0 suggests neutral initial stability (tender, slow to return, sensitive to loading changes).
• GM < 0 indicates initial instability (small heels tend to increase rather than self-correct).

This buoyancy-stability calculator estimates GM from three vessel dimensions:

• Beam, B (m): the vessel’s breadth at the waterline region relevant to initial stability (often approximated by maximum beam for simple checks).
• Center of Gravity above Keel, KG (m): vertical position of the vessel’s total weight (hull + machinery + payload + fluids) measured up from the keel/baseline.
• Center of Buoyancy above Keel, KB (m): vertical position of the centroid of displaced volume (the buoyant force line of action) measured up from the same reference.

Datum consistency matters: KG and KB must be measured from the same vertical reference (typically the keel/baseline). If you mix reference points (e.g., KG from the waterline and KB from the keel), the computed GM will be meaningless.

Core formulas for the initial transverse GM estimate

For initial transverse vessel stability, metacentric height is commonly written as:

GM = KM − KG, where KM = KB + BM.

The term BM (metacentric radius) depends on waterplane geometry and displacement:

• BM = I_T / ∇
• I_T = transverse second moment of area of the waterplane about the centerline
• ∇ = displaced volume (volume of water displaced)

This calculator uses a simplified “boxy waterplane” approximation for a quick initial-stability check when detailed hydrostatics are unavailable. With a rectangular waterplane and a coarse relationship between beam and draft, BM can be approximated as proportional to beam. A frequently used back-of-envelope approximation is:

BM ≈ B² / (12·d) and with a crude draft estimate d ≈ B/2, this reduces to BM ≈ B/6.

Accordingly, this calculator’s simplified vessel GM estimate is:

GM ≈ (KB + B/6) − KG

MathML for the GM calculation

Units: If B, KG, and KB are in meters, the estimated GM is in meters.

How to interpret the vessel GM result

A vessel’s GM is an initial stability indicator. It describes the slope of the righting-arm curve near zero heel and is most meaningful at small angles (often the first few degrees). A larger positive GM typically means the vessel is “stiffer” (returns upright more strongly for small angles), while a small positive GM indicates a “tender” vessel that may feel slow and roll more deeply.

However, a suitable GM is not a universal number. The same GM might be acceptable for one vessel type and uncomfortable or unsafe for another depending on:

• Vessel type (workboat, sailboat, barge, ferry, yacht)
• Loading condition (cargo height, passengers, tanks, deck loads)
• Sea state and operational profile
• Free-surface effects in slack tanks (can materially reduce effective GM)
• Downflooding points and reserve buoyancy (large-angle behavior)

Worked example: small-craft initial GM estimate

For a small craft, suppose the following values are estimated or measured from the keel/baseline:

• Beam B = 3.0 m
• Center of buoyancy KB = 0.60 m
• Center of gravity KG = 1.20 m

1. Compute approximate BM: BM ≈ B/6 = 3.0/6 = 0.50 m
2. Compute KM: KM = KB + BM = 0.60 + 0.50 = 1.10 m
3. Compute GM: GM = KM − KG = 1.10 − 1.20 = −0.10 m

Interpretation: A negative GM in this simplified vessel check suggests the craft may be initially unstable in that loading condition, or that the simplified geometry does not represent its actual waterplane and displacement. Treat this as a red flag: verify KG and KB, confirm the reference datum, consider tank free-surface, and obtain proper hydrostatic and stability analysis before operation.

Quick comparison table for initial vessel GM

This table gives qualitative context for a vessel’s computed GM at small heel angles. It is not a safety standard and cannot declare a vessel “safe” or “unsafe.”

Buoyancy-stability assumptions and limitations

• For vessel GM, small-angle only: GM characterizes initial stability near upright. It does not describe large-angle righting energy, range of positive stability, or capsize resistance.
• Simplified hull geometry: The BM estimate uses a coarse approximation (boxy/rectangular waterplane and a rough draft relationship). Real waterplanes (flare, tumblehome, chines, round bilges) can change I_T, ∇, and thus BM materially.
• No displacement or waterplane input: Because displacement volume ∇ and waterplane inertia I_T are not entered, the tool cannot compute BM from first principles for your exact condition.
• Free-surface effects not included: Partly filled tanks can reduce effective GM (sometimes dramatically). This calculator does not account for free-surface moments.
• No downflooding/edge immersion check: Deck edge immersion, openings, and downflooding points can dominate real-world safety but are outside the scope here.
• Static estimate: Waves, wind heel, turning, passenger motion, and dynamic stability effects are not modeled.
• Datum consistency required: KG and KB must be measured from the same baseline (typically keel). If unsure, stop and reconcile references before trusting any output.

Tips to improve vessel GM inputs

• For a more reliable vessel GM estimate, use published KM hydrostatics (or BM and KB) for the relevant displacement and trim instead of relying on the B/6 approximation.
• For KG, ensure you are using the current loading condition (fuel, water, cargo, passengers) and a consistent vertical datum.
• If tanks are slack, consider doing a separate free-surface correction with proper tank geometry.

Vessel stability references for deeper study

• Eugene A. Comstock (ed.), Principles of Naval Architecture, SNAME.
• Edward V. Lewis (ed.), Principles of Naval Architecture: Stability and Strength, SNAME.