Hydrostatics: Buoyancy & Stability

Hydrostatics: Buoyancy & Stability

$Learning Objectives$

Apply Archimedes' principle to fully submerged, partially submerged, and freely floating bodies.
Distinguish true weight, buoyant force, apparent weight, and net vertical force.
Determine displaced volume, draft, center of buoyancy, and combined center of gravity from force and moment balances.
Evaluate completely submerged stability from the relative locations of center of gravity and center of buoyancy.
Calculate waterplane inertia, metacentric radius, metacentric height, righting arm, and righting moment for initial floating-body stability.
Account for free-surface effects, ballast and load shifts, and recognize the limits of the small-angle metacentric approximation.
Apply buoyancy to civil-engineering flotation and groundwater-uplift checks.

$F_B$

Buoyant force is the resultant hydrostatic force exerted upward on an immersed body. Its magnitude equals the weight of the displaced fluid when the surrounding fluid is in hydrostatic equilibrium.

Archimedes' Principle

Relates buoyant force to the volume and specific weight of displaced fluid.

F_B=\rho_f gV_{\text{disp}}=\gamma_fV_{\text{disp}}

Variables

Symbol	Description	Unit
$F_B$	Buoyant force	N
$\rho_f$	Density of surrounding fluid	$kg/m^3$
$\gamma_f$	Specific weight of surrounding fluid	$N/m^3$
$V_{\text{disp}}$	Displaced-fluid volume	$m^3$
$g$	Gravitational acceleration	$m/s^2$

$B$

The center of buoyancy is the centroid of the displaced-fluid volume. The buoyant resultant acts vertically upward through this point in a hydrostatic fluid.

$G$

The center of gravity is the point through which the resultant gravitational weight of the body and its carried loads acts.

$Vertical Equilibrium and Buoyancy States$

A freely floating body at rest satisfies $F_B=W$ .
A fully submerged body is neutrally buoyant in translation when $F_B=W$ , which for a homogeneous body means its average density equals the surrounding-fluid density.
If $W>F_B$ for a fully submerged body, an additional upward support force of $W-F_B$ is required for static equilibrium.
If $F_B>W$ , a downward restraint is required to hold the body fully submerged.

Buoyancy free-body diagram

Weight and buoyancy act through different characteristic points.

Floating Displacement Requirement

Finds the displaced volume required for vertical equilibrium of a freely floating body.

V_{\text{disp}}=\frac{W}{\rho_f g}=\frac{W}{\gamma_f}

Variables

Symbol	Description	Unit
$W$	Total body and payload weight	N
$V_{\text{disp}}$	Required displaced volume	$m^3$
$\gamma_f$	Surrounding-fluid specific weight	$N/m^3$

Submerged Fraction for a Homogeneous Floating Body

Relates submerged fraction to the density ratio when the body has uniform density.

\frac{V_{\text{sub}}}{V_{\text{body}}}=\frac{\rho_{\text{body}}}{\rho_f}

Variables

Symbol	Description	Unit
$V_{\text{sub}}$	Submerged body volume	$m^3$
$V_{\text{body}}$	Total body volume	$m^3$
$\rho_{\text{body}}$	Average homogeneous body density	$kg/m^3$
$\rho_f$	Surrounding-fluid density	$kg/m^3$

$Apparent Weight$

Apparent weight is the support force indicated by a vertical cable or scale for a body in static immersion. For a body whose weight exceeds buoyancy, it equals true weight minus buoyant force.

Apparent Weight of a Supported Submerged Body

Computes the vertical support force for a body heavier than the displaced fluid.

W_{\text{app}}=W-F_B

Variables

Symbol	Description	Unit
$W_{\text{app}}$	Upward support force or scale reading	N
$W$	True gravitational weight	N
$F_B$	Buoyant force	N

$Interactive Buoyancy Exploration$

Vary body density, fluid density, and volume in the simulator. Compare the predicted displaced volume with the equilibrium condition $F_B=W$ and distinguish free floating from a restrained fully submerged state.

Buoyancy and Flotation Simulator

Learning objective: Relate fluid density, object density, displaced volume, buoyant force, and weight to floating, neutral, and sinking states.

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.

Combined Center of Gravity

Locates the center of gravity of a body and discrete carried loads from a common vertical datum.

KG=\frac{\sum W_i z_i}{\sum W_i}

Variables

Symbol	Description	Unit
$KG$	Height of combined center of gravity above the chosen datum	m
$W_i$	Component or load weight	N
$z_i$	Height of component center of gravity above the same datum	m

$Use One Datum for All Weight Moments$

Every $z_i$ , $KB$ , and $KG$ used in a stability calculation must be measured from the same datum. Mixing distances measured from the keel, deck, waterline, or another reference without conversion is a common source of sign and magnitude errors.

$Stability of a Completely Submerged Body$

For a completely submerged rigid body whose displaced volume does not change with a small rotation, stability is governed by the relative positions of $G$ and $B$ .

$Completely Submerged Stability$

Stable: $G$ lies below $B$ , so a small angular displacement creates a restoring couple.
Neutral: $G$ coincides with $B$ .
Unstable: $G$ lies above $B$ , so a small angular displacement creates an overturning couple.

This criterion is not the same as floating-body stability because a floating body's submerged geometry and center of buoyancy change as the body heels.

$M$

For a floating body subjected to a sufficiently small heel, the metacenter is the intersection of the shifted buoyancy vertical with the original upright centerline. It is a local geometric construct used for initial stability.

$BM$

Metacentric radius is the distance from the upright center of buoyancy to the initial metacenter for rotation about a specified heel axis.

Metacentric Radius

Relates waterplane geometry to displaced volume for initial floating-body stability.

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

Variables

Symbol	Description	Unit
$BM$	Metacentric radius	m
$I_{wp}$	Second moment of the waterplane area about the selected heel axis	$m^4$
$V_{\text{disp}}$	Displaced volume	$m^3$

$GM$

Metacentric height is the signed vertical distance from center of gravity to the initial metacenter. Positive $GM$ indicates an initial restoring tendency for small heel.

Metacentric Height from a Common Datum

Computes initial metacentric height using keel-based or other consistent vertical coordinates.

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

Metacenter and initial stability

Small heel shifts buoyancy and defines metacentric height.

$GB$

Use $GM=KB+BM-KG$ from one datum unless a signed $BG$ coordinate convention has been explicitly defined. Writing $GM=BM\pm GB$ without a sign convention is ambiguous and easily misapplied.

$GZ$

The righting arm is the perpendicular separation between the lines of action of weight and buoyancy after heel. Its sign determines whether their couple is restoring or overturning.

Initial Righting Arm and Righting Moment

Approximates the restoring lever and moment for small heel when the metacentric construction remains valid.

GZ\approx GM\sin\phi

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

Variables

Symbol	Description	Unit
$GZ$	Signed righting arm	m
$\phi$	Heel angle	degrees or rad
$M_R$	Righting moment magnitude for positive GM	N·m
$W$	Displacement weight	N

$Initial Stability Criterion$

$GM>0$ : initially stable for sufficiently small heel.
$GM=0$ : neutral initial stability in the idealized model.
$GM<0$ : initially unstable.

A very large positive $GM$ is not automatically desirable because strong restoring stiffness can produce rapid, high-acceleration rolling.

Righting arm at small heel

Separated force lines create a restoring moment.

$Interactive 3D Floating Body & Metacentric Stability$

Manipulate the heel angle $\phi$ , beam width, draft, and vertical center of gravity $KG$ in 3D. Watch how the center of buoyancy shifts ( $B \to B'$ ), the metacenter $M$ is formed, and the righting arm $GZ$ creates restoring vs overturning moments before capsize occurs.

Buoyancy & Metacentric Stability 3D

Interactive 3D vessel roll stability, shifting Center of Buoyancy ( $B \to B'$ ), Metacenter ( $M$ ), righting arm ( $GZ$ ), and capsizing dynamics.

Loading 3D Buoyancy & Metacentric Stability Simulation…

Metacenter & GZ ArmGravity & Buoyant Vectors

Vessel & Loading Geometry

Heel Angle (

\theta

)12°

Center of Gravity (

KG

)0.85 m

Hull Beam (

B

)2.40 m

Submerged Draft (

T

)0.90 m

Metacentric Height

\overline{GM}

0.133 m

Righting Arm

GZ

2.8 cm

Metacentric Radius

BM

0.533 m

Metacenter Height

KM

0.983 m

Righting Moment2.9 kN·m

Stability StatusSTABLE EQUILIBRIUM

$Metacentric Relations Are Small-Angle Relations$

The approximation $GZ\approx GM\sin\phi$ is intended for initial stability near the upright condition. At larger heel, the location of $M$ is no longer treated as fixed and the full nonlinear righting-arm curve, freeboard, deck-edge immersion, downflooding openings, shifting loads, and other operational constraints become important.

Heel from a Small Transverse Load Shift

Relates a transverse load shift to the equilibrium heel angle in the initial-stability approximation.

w x=W\,GM\tan\phi

Variables

Symbol	Description	Unit
$w$	Shifted load weight	N
$x$	Horizontal distance through which the load is shifted	m
$W$	Total displacement weight	N
$GM$	Corrected initial metacentric height	m
$\phi$	Resulting small equilibrium heel angle	degrees or rad

$Free-Surface Effect$

Free-surface effect is the reduction in effective initial stability caused by liquid shifting across the free surface of a partially filled internal compartment as the body heels.

Free-Surface Correction

Represents the destabilizing free-surface moment as a virtual rise in center of gravity.

FSC=\frac{\sum \rho_i I_{fs,i}}{\rho_dV_{\text{disp}}}

GM_{\text{corrected}}=GM_{\text{solid}}-FSC

Variables

Symbol	Description	Unit
$FSC$	Free-surface correction	m
$\rho_i$	Density of liquid in compartment i	$kg/m^3$
$I_{fs,i}$	Second moment of each internal free surface about the heel axis	$m^4$
$\rho_d$	Density of displaced external fluid	$kg/m^3$
$V_{\text{disp}}$	External displaced volume	$m^3$

Free-surface effect

Liquid shift in a partly filled tank reduces stability.

$Why Wide Partially Filled Compartments Are Critical$

For transverse roll of a rectangular internal free surface, the breadth normal to the heel axis enters the second moment cubed. A wide shallow tank can therefore create a large free-surface penalty. Longitudinal subdivision into narrower cells substantially reduces the summed free-surface inertia.

$Ballast and Load Placement$

Adding ballast changes both total displacement and the combined center of gravity. Low ballast usually lowers $KG$ and can increase $GM$ , but the added weight also increases displaced volume and may change draft, $KB$ , and $BM$ . A complete ballast calculation therefore recomputes the floating geometry rather than changing $KG$ alone.

Approximate Small-Roll Period

Estimates undamped small-roll period using mass radius of gyration and corrected metacentric height.

T\approx2\pi\sqrt{\frac{k^2}{gGM}}

Variables

Symbol	Description	Unit
$T$	Approximate natural roll period	s
$k$	Mass radius of gyration about the roll axis	m
$GM$	Positive corrected initial metacentric height	m
$g$	Gravitational acceleration	$m/s^2$

$Roll-Period Formula Is an Idealized Dynamic Estimate$

The expression neglects viscous damping, added water mass, waves, moorings, sloshing dynamics, and coupling with other motions. It is useful for instructional comparison of stiffness, not a complete dynamic-response model.

$Flotation or Groundwater Uplift Check$

A flotation check compares upward buoyancy or hydrostatic uplift with the available downward resisting weight and other justified restraints for a structure that could lift, float, or lose foundation contact.

Simple Uplift Safety Ratio

Compares modeled downward resistance with upward buoyancy for a specified load case.

FS_{\text{uplift}}=\frac{R_{\downarrow}}{F_B}

Variables

Symbol	Description	Unit
$FS_{\text{uplift}}$	Uplift safety ratio for the stated model	dimensionless
$R_{\downarrow}$	Total justified downward resistance	N
$F_B$	Upward buoyant or hydrostatic resultant	N

$Uplift Acceptance Criteria Are Project-Specific$

The equilibrium ratio alone does not establish design acceptability. Required safety factors, groundwater levels, anchorage contribution, soil cover, drainage, and load combinations must come from the governing design basis and project conditions.

$Floating-Body Initial-Stability Workflow$

STEP-BY-STEP

Sum all weights and compute $KG$ from a common datum.
Enforce $W=\gamma_fV_{\text{disp}}$ to determine displaced volume and draft.
Locate $B$ as the centroid of the submerged volume and determine $KB$ .
Compute $I_{wp}$ about the actual heel axis and then $BM=I_{wp}/V_{\text{disp}}$ .
Compute solid-loading $GM=KB+BM-KG$ .
Subtract applicable free-surface corrections to obtain corrected $GM$ .
Use small-angle righting or load-shift relations only while their geometric assumptions remain valid.
For design, separately check freeboard, openings, large-angle stability, operational loads, and the governing acceptance criteria.

$Civil Engineering Applications$

Floating and towing of precast caissons, pontoons, docks, and work platforms.
Pipeline flotation during installation or high groundwater.
Empty buried tanks, treatment units, basements, and slabs subjected to groundwater uplift.
Ballasting sequences for launching, immersion, and placement of large structures.
Temporary floating breakwaters, cofferdam units, and construction platforms.

Key Takeaways

Buoyant force equals the weight of displaced fluid and acts through the center of buoyancy.
Freely floating bodies satisfy $F_B=W$ ; apparent weight is a support-force concept, not a change in true mass.
Completely submerged rotational stability depends on the relative positions of $G$ and $B$ .
Initial floating stability depends on waterplane geometry through $BM=I_{wp}/V_{\text{disp}}$ and on load placement through $KG$ .
Use one datum for $KB$ , $KG$ , and $GM$ to avoid sign ambiguity.
Free-surface effects reduce effective $GM$ , and wide partially filled compartments can be especially destabilizing.
Ballast changes both center of gravity and displacement geometry, so draft and metacentric quantities should be recomputed.
Righting-arm, load-shift, and roll-period relations are local small-angle approximations, not substitutes for full large-angle or dynamic stability analysis.
Civil-engineering uplift checks compare buoyancy with justified resistance under a specified load case; acceptance criteria are not universal.

Previous TopicHydrostatics: Forces on Surfaces - Examples & Applications

Quiz Me

Next TopicHydrostatics: Buoyancy & Stability - Examples & Applications

Prev Next

Quiz Me

Learning Objectives

Buoyant Force, FBF_BFB​

Archimedes' Principle

Center of Buoyancy, BBB

Center of Gravity, GGG

Vertical Equilibrium and Buoyancy States

Floating Displacement Requirement

Submerged Fraction for a Homogeneous Floating Body

Apparent Weight

Apparent Weight of a Supported Submerged Body

Interactive Buoyancy Exploration

Buoyancy and Flotation Simulator

What this teaches

Combined Center of Gravity

Use One Datum for All Weight Moments

Stability of a Completely Submerged Body

Completely Submerged Stability

Metacenter, MMM

Metacentric Radius, BMBMBM

Metacentric Radius

Metacentric Height, GMGMGM

Metacentric Height from a Common Datum

Avoid an Undefined Plus-or-Minus GBGBGB Convention

Righting Arm, GZGZGZ

Initial Righting Arm and Righting Moment

Initial Stability Criterion

Interactive 3D Floating Body & Metacentric Stability

Buoyancy & Metacentric Stability 3D

Vessel & Loading Geometry

Metacentric Relations Are Small-Angle Relations

Heel from a Small Transverse Load Shift

Free-Surface Effect

Free-Surface Correction

Why Wide Partially Filled Compartments Are Critical

Ballast and Load Placement

Approximate Small-Roll Period

Roll-Period Formula Is an Idealized Dynamic Estimate

Flotation or Groundwater Uplift Check

Simple Uplift Safety Ratio

Uplift Acceptance Criteria Are Project-Specific

Floating-Body Initial-Stability Workflow

Civil Engineering Applications

$Learning Objectives$

$F_B$

$B$

$G$

$Vertical Equilibrium and Buoyancy States$

$Apparent Weight$

$Interactive Buoyancy Exploration$

$Use One Datum for All Weight Moments$

$Stability of a Completely Submerged Body$

$Completely Submerged Stability$

$M$

$BM$

$GM$

$GB$

$GZ$

$Initial Stability Criterion$

$Interactive 3D Floating Body & Metacentric Stability$

$Metacentric Relations Are Small-Angle Relations$

$Free-Surface Effect$

$Why Wide Partially Filled Compartments Are Critical$

$Ballast and Load Placement$

$Roll-Period Formula Is an Idealized Dynamic Estimate$

$Flotation or Groundwater Uplift Check$

$Uplift Acceptance Criteria Are Project-Specific$

$Floating-Body Initial-Stability Workflow$

$Civil Engineering Applications$