Metacentric Buoyancy Stability

Distinguish floating equilibrium from stability using the metacenter and restoring torque.

Metacentric Buoyancy Stability

The stability of a floating object is determined by the relative positions of its center of gravity (G), center of buoyancy (B), and the metacenter (M). When the object heels at a small angle, the center of buoyancy shifts to a new position. The metacenter is the intersection of the new vertical line of buoyant force and the original centerline of the object.

Worked Example

Consider a rectangular barge with length L = 4 m, beam B = 2 m, and draft d = 0.5 m. The density of water is 1000 kg/m³, and gravity g is 9.81 m/s². The center of gravity (KG) is 0.5 m from the keel. Let's calculate its initial stability.

First, calculate the height of the center of buoyancy (KB):
KB = d / 2 = 0.5 / 2 = 0.25 m
Next, calculate the metacentric radius (BM):
BM = B² / (12 * d) = 2² / (12 * 0.5) = 4 / 6 = 0.6666666666666666 m
Then, calculate the metacentric height (GM):
GM = KB + BM - KG = 0.25 + 0.6666666666666666 - 0.5 = 0.41666666666666663 m
Since GM > 0, the barge is initially stable. For a heel angle of 5 degrees (0.087266 radians), the righting arm GZ is:
GZ = GM * sin(5°) ≈ 0.41666666666666663 * 0.0871557 = 0.03631489281152423 m
The displaced volume V = L * B * d = 4 * 2 * 0.5 = 4 m³. The restoring torque is:
Torque = -1000 * 9.81 * 4 * 0.03631489281152423 = -1424.996393924211 N·m
Expected result: The restoring torque is -1424.996393924211 N·m.

Experiment

In the interactive module, you can alter the beam, draft, center of gravity, and heel angle to see how these factors impact stability. You can try this yourself in the Metacentric Buoyancy Stability module.

Set the Beam to 2, Draft to 0.5, and Center of gravity to 0.5.
Set the Heel degrees to 5.
Observe that the Initial stability is Stable, and restoring torque is -1424.996393924211 N·m.
Increase the Center of gravity to 1. Observe that the Metacentric height (GM) becomes negative and Initial stability changes to Unstable.

Limitations

This model uses the small-angle approximation, which assumes that the metacenter M remains fixed on the centerline for small angles of heel (typically up to about 5-10 degrees). For larger angles, the metacenter moves, and this simple formula for the righting arm (GZ = GM * sinθ) is no longer accurate. This analysis also does not account for dynamic effects, wave action, or flooding.

References

MIT OpenCourseWare: 2.016 Hydrodynamics (13.012), Fall 2005

This course provides comprehensive material on ship hydrostatics, including the derivation of metacentric height and stability criteria for floating bodies.

Metacentric Buoyancy Stability