Engineering & Technology · Marine, Ocean & Offshore Engineering
Buoyancy, hydrostatics and vessel stability
A floating object has enough buoyancy to balance its weight, yet it may roll away from an upright position. Vertical force balance and stability against a small rotation are different questions. Both depend on the actual loading and the submerged geometry.
Use Fluid Engineering, Civil & Structural Engineering and Materials Selection & Degradation for balances, structural response and environmental mechanisms. Physics fluid dynamics and continuum materials provide further preparation. No example supplies a marine operating procedure, stability certificate, mooring design, electrical architecture approval or environmental permission.
Back to the chapterAfter reading, you can
Compute buoyant equilibrium and interpret initial metacentric stability without confusing flotation with complete vessel stability.
Buoyancy is a pressure resultant
In a stationary fluid of uniform density, pressure increases with depth. Integrating pressure over a closed wetted body gives an upward buoyant force equal to the weight of displaced water. For a freely floating body with no other vertical support, equilibrium requires mg=ρg∇, where ∇ is submerged volume. The centre of buoyancy B is the centroid of that volume.
Draft measures immersion relative to the vessel; freeboard describes the distance from the water surface to a specified deck or opening level. A displaced volume calculation alone does not establish freeboard, watertight integrity or what happens after damage. Water density changes also alter the immersion needed to support an unchanged mass.
Waterplane geometry affects the initial restoring moment
As a surface-piercing hull heels slightly, the submerged shape and B shift. The initial transverse metacentre M describes this geometric change. With a consistent vertical reference, GM=KB+BM−KG, where G is centre of gravity and BM=Iwp/∇. Here Iwp is the waterplane area’s second moment about its longitudinal centreline, not the hull material’s structural bending inertia.
Positive GM gives an initial restoring tendency in this model; negative GM gives an overturning tendency near upright. For small heel θ, a metacentric approximation uses righting arm GZ≈GM sin θ and restoring-moment magnitude W GZ. This local approximation does not supply the complete righting-arm curve, a downflooding angle or a large-angle recovery guarantee.
Keep the geometry and loading explicit
Take an invented rectangular floating box, length 12 m, beam 6 m and upright draft 1 m. Supply uniform water density 1025 kg/m³. Submerged volume is 72 m³ and supported mass 73800 kg. The buoyancy centroid lies 0.50 m above its flat keel. Its rectangular waterplane gives Iwp=12×6³/12=216 m⁴, so BM=3.0 m.
If the supplied KG is 2.0 m, GM=0.5+3−2=1.5 m. Raising G to 4.0 m with the same mass and upright volume gives GM=−0.5 m. Both cases satisfy the same vertical buoyancy balance but have opposite initial rotational tendencies. The figure shows only the declared small-angle approximation around upright.
Calculate one small-angle restoring moment
With g=9.81 m/s², the original weight is 723978 N. At a supplied five-degree heel and GM=1.5 m, the approximate arm is 1.5 sin 5°≈0.13073 m, giving restoring moment about 94.65 kN·m. Its direction opposes the heel. In the negative-GM case the local tendency instead reinforces the displacement.
Actual loading changes, slack-liquid free surfaces, flooding and hull geometry can change stability. Strong initial restoring action also does not guarantee comfortable motion. The hypothetical box has no complete deck, opening, damage or wave-response assessment; the arithmetic is neither a loading instruction nor an acceptance test for a real vessel.
Try the reasoning
Does unchanged total mass guarantee unchanged initial stability?
Show the answer and reasoning
No. Its vertical distribution can change KG, while liquid free surfaces and other changes can alter the effective restoring behavior.
Which second moment appears in BM=Iwp/∇?
Show the answer and reasoning
The second moment of the waterplane area about the relevant horizontal rotation axis, not the material cross-section used for structural EI.
Does positive GM prove large-angle or damaged stability?
Show the answer and reasoning
No. It describes an initial tendency near the stated equilibrium; complete stability needs the relevant geometry, loading and damage conditions.
What to carry forward
Close the buoyancy balance, then examine the loading-dependent restoring behavior and the range over which the stability model applies.
References
Teaching resources and research supporting this lesson’s explanations and examples.