Boat Initial Stability at Small Heel Angles

Initial stability describes a boat’s ability to resist small angles of heel about its upright equilibrium position. It refers to the vessel’s behavior at small heel angles, typically up to about 5–10 degrees, where changes in the geometry of the submerged volume are small and linear approximations are valid. When the boat heels slightly, the center of buoyancy B shifts laterally relative to the center of gravity G, creating a righting arm GZ. For small angles of heel, this righting arm is approximately proportional to the heel angle and is governed primarily by the metacentric height (GM). A larger GM results in a larger righting moment for a given small heel angle, corresponding to a “stiffer” vessel. Initial stability is an important indicator of how stable a boat feels in normal operating conditions, such as crew movement, small waves, or light wind. However, it does not describe the vessel’s behavior at large angles of heel and is not sufficient on its own to assess overall safety. For that purpose, the full righting-arm (GZ) curve and dynamic stability characteristics must be considered.
Diagram showing metacentric height GM and righting arm at small heel angles

Initial Stability of a Boat

  1. K — Keel
    Point K is the keel reference point. It is commonly used as the vertical datum from which other vertical distances are measured, such as KG, KB, and KM. While K itself has no direct influence on stability, it provides a consistent reference for defining the vessel’s geometry.
  2. W — Waterline
    W denotes the waterline corresponding to the given loading condition. The geometry of the waterline determines the shape of the submerged volume and strongly influences the transverse metacentric radius and initial stability.
  3. B — Center of Buoyancy (upright)
    Point B is the center of buoyancy in the upright condition. It represents the centroid of the underwater volume and is the point through which the buoyant force acts when the boat is upright.
  4. Bφ — Center of Buoyancy When Heeled
    When the boat heels by a small angle φ, the shape of the submerged volume changes asymmetrically. As a result, the center of buoyancy moves from B to Bφ. This lateral shift of the buoyancy force is the fundamental cause of the restoring (righting) moment.
  5. G — Center of Gravity
    Point G is the center of gravity of the boat. It is determined by the mass distribution of the hull, structure, equipment, payload, and crew. For small angles of heel, G is assumed to remain fixed relative to the hull.
  6. Z — Righting Arm Point
    Point Z is the foot of the perpendicular drawn from G to the line of action of the buoyancy force acting through Bφ. The horizontal distance GZ is the righting arm.
  7. M — Metacenter
    Point M is the metacenter, defined as the intersection of the buoyancy force line (through Bφ) with the vessel’s centerline for an infinitesimal angle of heel. For small heel angles, M may be considered fixed. The relative position of M with respect to G determines the initial stability.
  8. Metacentric Height (GM)
    The vertical distance between the center of gravity (G) and the metacenter (M). The metacenter determines the restoring force when the boat heels slightly. A positive GM indicates a restoring moment, causing the boat to return upright.
  9. Righting Arm (GZ)
    The horizontal distance between the lines of action of gravity and buoyancy when the boat is heeled. The GZ curve plotted against heel angle shows that, for small angles (up to 5–10°), GZ is nearly linear and proportional to GM.
  10. Initial Stability
    Determined by the slope of the GZ curve at small heel angles. Larger GM → greater stability. At small heel angles, a positive restoring moment returns the boat upright.
  11. Summary of Interactions
    Gravity at G acts downward, and buoyancy at B acts upward. When the boat heels, the center of buoyancy shifts, creating a horizontal distance (GZ) that generates a restoring moment. Positive moment at small angles demonstrates initial stability, with GM as the key indicator.

Meaning of Distances and Stability Measures

GM — Metacentric Height

The distance GM is the vertical separation between the center of gravity G and the metacenter M. It is the primary quantitative measure of initial stability.

A larger GM produces a stronger restoring moment for small angles of heel, resulting in a “stiffer” vessel. A smaller GM corresponds to a “softer” motion but reduced initial stability.

GZ — Righting Arm

The distance GZ is the horizontal lever arm between the lines of action of buoyancy and weight. It directly determines the righting moment, calculated as:

Righting Moment = Displacement × GZ

For small angles of heel, GZ is approximately proportional to the heel angle φ and to GM, and may be expressed as:

GZ ≈ GM × sin(φ)

This linear relationship is valid only in the range of small angles, where initial stability theory applies.

KB, KG, KM (implicit distances)

Although not always shown explicitly, the following vertical distances are commonly used:

They are related by the fundamental relation: GM = KM − KG

Interpretation

The diagram demonstrates how a small heel causes the buoyancy force to shift laterally while the weight force remains fixed, creating a restoring couple. Initial stability analysis focuses exclusively on this geometric mechanism and is valid only for small angles of heel. Behavior at larger angles requires consideration of the full righting-arm curve and nonlinear effects.


Examples of Boats with Good and Poor Stability

This Boat Has Poor Initial Stability

Boat with poor initial stability
Waterline level (mm) Righting Arm GZ (mm) Metacentric Height GM (mm) Volume below the waterline (m 3)
100 -3.08 -88.17 0.069
200 -0.96 -27.47 0.261
300 -1.10 -31.55 0.549

The green line indicates the water level. At 100 mm, the boat will clearly fall onto its side; the same applies at 200 mm. Even at 300 mm, the boat is unstable because the righting arm GZ is negative.
Conclusion: A boat with this hull shape is not stable. It could be made stable, but it would require substantial ballast or a heavy keel.

This Boat Has Good Initial Stability

Boat with good initial stability
Waterline level (mm) Righting Arm GZ (mm) Metacentric Height GM (mm) Volume below the waterline (m 3)
100 21.76 623.48 0.219
200 12.24 350.75 0.636
300 8.72 249.87 1.155

This boat has positive values for the righting arm GZ and metacentric height GM, indicating very good initial stability.
Conclusion: A boat with this hull shape is very stable.
The calculations were performed with BoatCAD.

Reverse Engineering of a Mediterranean and Black Sea Boat: Initial Stability Analysis

Many boats in the Mediterranean and Black Sea have a similar hull shape, but no two are identical. These boats are over 40 years old. I do not know any young craftsman who can build a new one. I believe they were all built without blueprints, which is why they are not identical. I took measurements and created a blueprint. The blueprint for this boat is available in the examples under the name PM462.
Mediterranean boat initial stability analysis

These Are the Initial Stability Calculations for This Boat


LWL5483.03mm
Volume below the waterline2.277m3
Waterplane area7.433m2
Hull surface area below the waterline9.261 m2
Total area7.806m 2
Initial stability
Angle 2°
BM354.37mm
GM59.86mm
GZ2.09mm
Angle 4°
BM357.10mm
GM62.59mm
GZ 4.37mm
Conclusion: The boat will not capsize easily, but it will have a slow, “soft” righting response and will heel noticeably when a person or cargo is moved. The boat is narrow (1.72 m is a small beam for a 6 m hull). It has a high center of gravity. Reference values for comparison: For boats of approximately 6 m length: GM ≈ 50–80 mm → soft, comfortable, but sensitive; GM ≈ 100–200 mm → normal stiffness; GM > 250 mm → very stiff, harsh response in waves. This is at the lower end of the range.
There is also one difference. When I measured the boat, it was in a horizontal position. In practice, its position changes when it is in the water. The bow is higher than the stern. This will affect the results under actual operating conditions.
The center of the boat is at section 7. The engine is usually located over section 6 or between sections 5 and 6.

This Is the Boat: a Classic Double-ender


Mediterranean boat initial stability analysis Mediterranean boat initial stability analysis

The sea is beautiful even in January.