Hydrostatics: Buoyancy & Stability

Hydrostatics: Buoyancy & Stability

Learning Objectives

Apply Archimedes' principle to fully submerged, partially submerged, and floating bodies.
Distinguish weight, buoyant force, apparent weight, and net vertical force.
Locate the center of buoyancy and determine the draft or displaced volume required for flotation.
Evaluate the stability of completely submerged bodies from the relative positions of the center of gravity and center of buoyancy.
Calculate metacentric radius, metacentric height, righting arm, and righting moment for small-angle floating-body stability.
Account for free-surface effects in partially filled compartments and recognize the limits of the metacentric approximation.

Buoyancy is the resultant of the hydrostatic pressure distribution on an immersed body. In civil engineering it governs the flotation and installation of caissons, pontoons, floating docks, pipelines, tanks, gates, temporary works, and submerged foundations. Stability analysis determines whether a displaced body returns toward equilibrium, remains in its new position, or overturns.

Archimedes' Principle

A body wholly or partly immersed in a fluid experiences an upward force equal to the weight of the fluid displaced by the submerged portion of the body. The force acts through the centroid of the displaced fluid volume, called the center of buoyancy.

Buoyant Force

Calculates the hydrostatic resultant on a body from the displaced-fluid volume.

F_B = \rho_f g V_{disp} = \gamma_f V_{disp}

Variables

Symbol	Description	Unit
$F_B$	Buoyant force	N
$\rho_f$	Density of the surrounding fluid	$kg/m^3$
$g$	Acceleration due to gravity	$m/s^2$
$V_{disp}$	Displaced-fluid volume, equal to submerged body volume	$m^3$
$\gamma_f$	Specific weight of the surrounding fluid	$N/m^3$

Vertical Force States

Floating at rest: $F_B=W$ . Only enough volume is submerged to displace a fluid weight equal to the body weight.
Neutral buoyancy: For a fully submerged body, $F_B=W$ and the average body density equals the fluid density.
Sinking tendency: $W>F_B$ when the body is fully submerged; the remaining downward force is $W-F_B$ .
Rising tendency: $F_B>W$ for a restrained submerged body; the remaining upward force is $F_B-W$ .

Floating Displacement Requirement

Finds the displaced volume or submerged fraction of a freely floating body.

V_{disp}=\frac{W}{\rho_f g} \qquad \frac{V_{disp}}{V_{body}}=\frac{\rho_{body}}{\rho_f}

Variables

Symbol	Description	Unit
$W$	Body weight	N
$V_{body}$	Total body volume	$m^3$
$\rho_{body}$	Average body density	$kg/m^3$

Apparent Weight

For a fully submerged body supported by a cable or scale,

W_{app}=W-F_B

This is a force balance, not a change in the body's true mass or gravitational weight.

Interactive Simulation

Vary the body density, fluid density, and body volume. The simulator distinguishes floating, neutral, and sinking states and reports the displaced volume required by equilibrium.

Buoyancy and Flotation Simulator

Fluid Density (

\rho_f

): 1000 kg/m³

Object Density (

\rho_o

): 600 kg/m³

Object Volume (

V

): 1.0 m³

Weight (

W

):5.89 kN

Buoyant Force (

F_B

):5.89 kN

Density Ratio (

\rho_o/\rho_f

):0.600

Status:FLOATING

Submerged volume:0.600 m³ (60.0%)

Fluid surface

ρₒ/ρf
0.60

F_B

W

What this teaches

A floating body displaces enough fluid for buoyant force to equal its weight. Neutral buoyancy occurs when object and fluid densities are equal. A denser object cannot obtain enough buoyant force even when fully submerged, so it sinks.

Center of Buoyancy, $B$

The centroid of the displaced-fluid volume. The resultant buoyant force acts vertically upward through this point.

Center of Gravity, $G$

The point through which the resultant body weight acts vertically downward. Moving equipment, ballast, stored liquids, or suspended loads can shift $G$ and change stability.

Stability of a Completely Submerged Body

For a completely submerged rigid body, the displaced volume and center of buoyancy do not change appreciably for a small rotation.

Stable: $G$ is below $B$ . A small rotation creates a restoring couple.
Neutral: $G$ coincides with $B$ .
Unstable: $G$ is above $B$ . A small rotation creates an overturning couple.

This criterion differs from floating-body stability because a floating body's center of buoyancy shifts when its waterplane shape changes during heel.

Metacenter of a Floating Body

When a floating body heels through a small angle, the submerged shape changes and the center of buoyancy moves from $B$ to $B'$ . The new buoyancy vertical intersects the original centerline at the metacenter, $M$ . The distance $GM$ is the initial metacentric height.

Metacentric Radius

Relates the waterplane second moment of area to the displaced volume.

BM=\frac{I_{wp}}{V_{disp}}

Variables

Symbol	Description	Unit
$BM$	Metacentric radius from center of buoyancy to metacenter	m
$I_{wp}$	Second moment of the waterplane area about the heel axis	$m^4$
$V_{disp}$	Displaced volume	$m^3$

Metacentric Height from a Common Datum

Avoids ambiguous plus/minus signs by locating all points from the same vertical datum, commonly the keel.

GM=KM-KG=KB+BM-KG

Variables

Symbol	Description	Unit
$GM$	Initial metacentric height	m
$KB$	Height of center of buoyancy above the datum	m
$BM$	Metacentric radius	m
$KG$	Height of center of gravity above the same datum	m

Do Not Use an Undefined $\pm GB$ Sign

Writing $GM=BM\pm GB$ without a signed coordinate convention is error-prone. Use $GM=KB+BM-KG$ from a common datum, or explicitly define the sign of $BG$ before using $GM=BM-BG$ .

Initial Stability Criterion

Stable: $GM>0$ and $M$ lies above $G$ .
Neutral: $GM=0$ .
Unstable: $GM<0$ and $M$ lies below $G$ .

A very large positive $GM$ produces a strong restoring moment but can cause rapid, uncomfortable, or structurally severe rolling. Stability therefore is not simply “the larger, the better.”

Small-Angle Righting Arm and Moment

Calculates the restoring lever and couple for a small heel angle.

GZ \approx GM\sin\phi

M_R=W\,GZ\approx W\,GM\sin\phi

Variables

Symbol	Description	Unit
$GZ$	Righting arm	m
$\phi$	Heel angle	degrees or rad
$M_R$	Restoring or righting moment	$N\cdot m$

Small-Angle Limit

The metacentric relation is an initial-stability approximation, commonly used for heel angles below roughly 10° to 15°. At larger angles, deck-edge immersion, freeboard, openings, shifting cargo, and the nonlinear righting-arm curve $GZ(\phi)$ must be evaluated directly.

Free-Surface Effect

A partially filled tank or compartment develops a moving liquid free surface when the body heels. The liquid shifts toward the low side, creating an overturning moment that behaves like a virtual rise of the body's center of gravity and reduces effective metacentric height.

Free-Surface Correction

Estimates the reduction in metacentric height caused by partially filled compartments.

FSC=\frac{\sum \rho_i I_{fs,i}}{\rho_d V_{disp}}

GM_{corrected}=GM_{solid}-FSC

Variables

Symbol	Description	Unit
$FSC$	Free-surface correction or virtual rise of G	m
$\rho_i$	Density of liquid in compartment i	$kg/m^3$
$I_{fs,i}$	Second moment of the compartment free surface about the heel axis	$m^4$
$\rho_d$	Density of the displaced surrounding fluid	$kg/m^3$

Why Wide, Shallow Tanks Are Critical

The free-surface correction depends on the second moment of the free-surface area. A wide compartment can cause a large stability penalty even when it contains relatively little liquid. Subdivision with longitudinal bulkheads greatly reduces $I_{fs}$ and the correction.

Approximate Small-Roll Period

Relates roll period to radius of gyration and corrected metacentric height for small undamped oscillations.

T\approx 2\pi\sqrt{\frac{k^2}{g\,GM}}

Variables

Symbol	Description	Unit
$T$	Natural roll period	s
$k$	Mass radius of gyration about the roll axis	m
$GM$	Corrected metacentric height	m

Real Motions Are Damped and Coupled

The roll-period expression neglects viscous damping, added mass, waves, mooring stiffness, and coupling with pitch, heave, or structural flexibility. It is an instructional estimate, not a complete dynamic stability analysis.

Floating-Body Stability Workflow

STEP-BY-STEP

Determine total weight and locate $G$ from component weights and moments.
Enforce vertical equilibrium, $W=\rho_f gV_{disp}$ , to obtain displaced volume and draft.
Locate $B$ as the centroid of the submerged volume and determine $KB$ .
Calculate the waterplane second moment $I_{wp}$ about the expected heel axis.
Compute $BM=I_{wp}/V_{disp}$ and $GM=KB+BM-KG$ .
Subtract all applicable free-surface corrections.
Check small-angle righting moment, operational loading cases, freeboard, openings, and large-angle criteria when required.

Civil Engineering Applications

Stability and towing draft of precast bridge or harbor caissons.
Flotation of pipelines during installation, flooding, or high groundwater.
Uplift checks for buried tanks, basements, slabs, and empty treatment units.
Stability of floating breakwaters, work platforms, pontoons, and temporary cofferdam units.
Ballasting sequences during launching, immersion, and placement of large concrete structures.

Key Takeaways

Buoyant force equals the weight of displaced fluid and acts through the center of buoyancy.
A freely floating body displaces exactly enough fluid for $F_B=W$ ; its submerged fraction follows the body-to-fluid density ratio only for a uniform body.
Fully submerged stability depends directly on the relative positions of $G$ and $B$ .
Floating-body initial stability requires the waterplane moment of inertia through $BM=I_{wp}/V_{disp}$ .
Use $GM=KB+BM-KG$ from a common datum to avoid sign ambiguity.
Partially filled compartments reduce stability through the free-surface correction and must be included before using righting-moment or roll-period formulas.
Metacentric formulas are small-angle tools; large-angle stability requires the complete righting-arm curve and operational checks.

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Learning Objectives

Archimedes' Principle

Buoyant Force

Vertical Force States

Floating Displacement Requirement

Apparent Weight

Interactive Simulation

Buoyancy and Flotation Simulator

What this teaches

Center of Buoyancy, BBB

Center of Gravity, GGG

Stability of a Completely Submerged Body

Metacenter of a Floating Body

Metacentric Radius

Metacentric Height from a Common Datum

Do Not Use an Undefined $\pm GB$ Sign

Initial Stability Criterion

Small-Angle Righting Arm and Moment

Small-Angle Limit

Free-Surface Effect

Free-Surface Correction

Why Wide, Shallow Tanks Are Critical

Approximate Small-Roll Period

Real Motions Are Damped and Coupled

Floating-Body Stability Workflow

Civil Engineering Applications

Center of Buoyancy, $B$

Center of Gravity, $G$