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” initial stability. 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 at heel
    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 gravity and buoyancy lines during heel. The GZ curve vs. 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, 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.

Heel Angle (°) Righting Arm GZ (m) Moment (kNm)
0 0.00 0.0
2 0.05 1.2
5 0.12 3.0
10 0.22 5.8

Reverse engineereing of Mediteranian - Black See boat. Initial Stability analyze.

There are a lot of boats in Mediteranian and Black see with similar shape like this. But there isn't 2 equal. This boats are over 40 years old. I don't know any yung craftsman how can build a new one. I guess all they are build without blueprint, this is the reason why they are not equal. I take the measurement and try to create a blueprint. The blueprint of this boat is available in examples with name PM462.
Mediteranian boat initial stability analyze

This is the calculations of initial stability of this boat.


LWL5483.03mm
Volume under waterline2.277m3
Waterline plain surface7.433m2
Surface boards under waterline9.261 m2
Surface all7.806m 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 1 difference. When I measure the boat it was in horizontal position. On reality when the boat is in the water it has some different. The front side is higer then stern. This will give some difference in the real situation.
The center of the boat is section 7. The engine is usually over section 6 or between 5 and 6.

This is the boat.


Mediteranian boat initial stability analyze Mediteranian boat initial stability analyze