Transverse Metacentric Height (GM)

Transverse Metacentric Height (GM) in Ship Stability

The transverse metacentric height, written as GM, is one of the most widely used indicators of a ship’s initial (small-angle) transverse stability. In simple terms, GM describes how strongly a vessel “wants” to return upright after a small heel. It is used in stability booklets, loading guidance, operational checks, and as a quick diagnostic value during design and trials.

This calculator provides two common engineering routes: Direct Hydrostatics (using KM & KG) and Waterplane Inertia & Displacement (using BM = I_T/∇), with an optional Free Surface Correction (FSC) to account for slack tanks.

What GM physically represents

When a vessel heels slightly, the underwater shape changes and the buoyancy force shifts sideways. The intersection between the buoyancy force line and the centerline at small heel angles defines the metacenter (M). The vertical distance between the vessel’s center of gravity (G) and M is GM:

GM = KM − KG

If GM is positive, the vessel has initial transverse stability and produces a restoring moment for small angles. If GM is negative, the vessel is initially unstable and will tend to heel further unless restrained.

Mode A: Direct hydrostatics (KM & KG)

The most common “operational” method uses hydrostatic data (from a stability booklet, loading computer, or hydrostatic tables) to read KM for the current draft/trim and combine it with the actual KG for the loading condition:

GM = KM − KG

This route is fast and practical because KM is usually available for many drafts and displacements, and KG can be tracked through loading calculations (weights, moments, consumables, ballast, etc.).

Mode B: Inertia & displacement route (BM = I_T/∇)

In design work (or when a hydrostatic KM table is not readily available), GM can be built from its components: KM = KB + BM. Here, KB is the vertical center of buoyancy above keel, and BM is the transverse metacentric radius determined by the waterplane geometry:

BM = I_T / ∇    and    KM = KB + BM

I_T is the second moment of area of the waterplane about the ship’s centerline (m⁴), and ∇ is displaced volume (m³). A wider waterplane and larger I_T generally increases BM, which tends to increase GM (for a fixed KG).

In this calculator, if ∇ is not entered and displacement Δ is available (in tonnes), a helper relation can be used: ∇ ≈ Δ / ρ (with ρ in t/m³). This is commonly used for quick checks at sea.

Free Surface Correction (FSC) & why it matters

Slack tanks (partially filled tanks) allow liquid to move as the ship heels, shifting the effective center of gravity and reducing stability. This effect is accounted for by the free surface correction:

GM_corrected = GM − FSC

Operationally, FSC is often computed from the sum of free surface moments:

FSC = ΣFSM / Δ

where ΣFSM is the total free surface moment (commonly in t·m) across all slack tanks and Δ is displacement (t). Even when “raw” GM looks healthy, a large FSC can reduce GM_corrected into a risky range. That’s why stability guidance typically emphasizes corrected GM.

Typical GM ranges & interpretation

• GM < 0 — initial transverse instability (unsafe without external restraint).
• GM slightly positive — tender ship; slower roll, larger angles for a given disturbance.
• GM moderate — generally desirable balance between stability and comfort.
• GM very large — stiff ship; quick/violent rolling, higher accelerations, comfort & lashing concerns.

“Good” GM is not a universal number; it depends on vessel type, size, loading condition, freeboard, intended service, and stability criteria. Use GM as a fast indicator, but always evaluate it together with righting-lever (GZ) characteristics, intact stability criteria, and operational limitations.

Common mistakes & best practices

• Mixing units: ensure I_T is in m⁴ and volume ∇ is in m³.
• Using uncorrected GM when slack tanks exist; always consider FSC where applicable.
• Using KM from the wrong draft/trim or wrong displacement row in hydrostatic tables.
• Assuming KB estimation rules apply universally; treat any KB estimate as a rough check only.
• Interpreting GM alone as “safe/unsafe” without considering GZ curve area & range of stability.

Related stability & load calculators

GM is one piece of intact stability. For a fuller picture, use it together with these tools:

• Free Surface Correction (FSC) – compute FSC from ΣFSM and displacement.
• GZ Curve Generator – evaluate righting lever across heel angles (range of stability & area criteria).
• Trim Calculator – check trim changes that affect hydrostatics and KM.
• MCT 1 cm – longitudinal stability metric used with trim and loading calculations.

Tip: If your GM changes unexpectedly between conditions, check KG (weight moments), slack tanks (FSC), and whether your KM/KB values match the correct displacement, draft, and trim.