StabilityStability & & Buoyancy Buoyancy by by

ABHISHEK KUMAR ABHISHEK KUMAR

ObjectivesObjectives• Principles of StabilityPrinciples of Stability• Archimedes PrincipleArchimedes Principle• Terminology of ship’s hydrostaticsTerminology of ship’s hydrostatics• Stability & moments -> staying uprightStability & moments -> staying upright• Metacenter, Center of Gravity, Center of Metacenter, Center of Gravity, Center of

Buoyancy, etc.Buoyancy, etc.• Stability curvesStability curves

Principles of StabilityPrinciples of Stability• Floating object is acted on by forces of gravity Floating object is acted on by forces of gravity

and forces of buoyancyand forces of buoyancy• Static equilibrium Static equilibrium ΣΣFFii = 0 = 0

• Three conditions of static equilibrium:Three conditions of static equilibrium:• Stable:Stable: return to same position if tipped return to same position if tipped

• Neutral:Neutral: when rotated, will come to rest in any when rotated, will come to rest in any positionposition

• Unstable:Unstable: will come to rest in new position if force will come to rest in new position if force acts on it acts on it

Archimedes PrincipleArchimedes Principle• Law: a body floating or submerged in a Law: a body floating or submerged in a

fluid is buoyed up by a force equal to the fluid is buoyed up by a force equal to the weight of the water it displacesweight of the water it displaces

• Depth to which ship sinks depends on Depth to which ship sinks depends on density of water (density of water (ρρ = 1 ton/35ft = 1 ton/35ft33 seawater)seawater)

Archimedes PrincipleArchimedes Principle• Ship sinks until weight of water Ship sinks until weight of water

displaced by the underwater volume is displaced by the underwater volume is equal to the weight of the shipequal to the weight of the ship

• Forces of gravity: Forces of gravity: G = mG = mshipshipg =Wg =Wshipship

• Forces of buoyancy: Forces of buoyancy: B = B = ρρ waterwaterVVdisplaceddisplaced

WWship ship = = ρρwaterwaterVVdisplaceddisplaced

Archimedes PrincipleArchimedes Principle• Forces act everywhere on ship -> too Forces act everywhere on ship -> too

tough to analyzetough to analyze• Center of Gravity (G):Center of Gravity (G): all gravity forces all gravity forces

as one force acting downward through as one force acting downward through ship’s geometric centership’s geometric center

• Center of Buoyancy (B):Center of Buoyancy (B): all buoyancy all buoyancy forces as one force acting upward forces as one force acting upward through underwater geometric centerthrough underwater geometric center

Archimedes PrincipleArchimedes Principle• Center of Gravity (G):Center of Gravity (G):

• Changes position only by change/shift in Changes position only by change/shift in mass of shipmass of ship

• Does not change positionDoes not change position with movement of with movement of shipship

• Center of Buoyancy (B):Center of Buoyancy (B):• Changes positionChanges position with movement of ship -> with movement of ship ->

underwater geometric center movesunderwater geometric center moves• Also affected by displacementAlso affected by displacement

G

Hydrostatics TerminologyHydrostatics Terminology• Displacement:Displacement: total weight of ship = total total weight of ship = total

submerged volume of ship (measured in tons)submerged volume of ship (measured in tons)• Draft:Draft: vertical distance from waterline to keel at vertical distance from waterline to keel at

deepest point (measured in feet)deepest point (measured in feet)• Reserve Buoyancy:Reserve Buoyancy: volume of watertight portion volume of watertight portion

of ship above waterline (important factor in of ship above waterline (important factor in ship’s ability to survive flooding)ship’s ability to survive flooding)

• Freeboard:Freeboard: vertical distance from waterline to vertical distance from waterline to main deck (rough indication of reserve main deck (rough indication of reserve buoyancy)buoyancy)

Hydrostatics TerminologyHydrostatics Terminology

• As draft & displacement increase, As draft & displacement increase, freeboard and reserve buoyancy decreasefreeboard and reserve buoyancy decrease

MomentsMoments• Def’n: tendency of a force to produce Def’n: tendency of a force to produce

rotation or to move an object about an rotation or to move an object about an axisaxis• Distance between the force and axis of Distance between the force and axis of

rotation is the moment armrotation is the moment arm• Couple: two forces of equal magnitude in Couple: two forces of equal magnitude in

opposite and parallel directions, opposite and parallel directions, separated by a perpendicular distanceseparated by a perpendicular distance• G and B are a coupleG and B are a couple

MomentsMoments• Depending on location of G Depending on location of G

and B, two types of and B, two types of moments:moments:• Righting moment:Righting moment: tends to tends to

return ship to upright positionreturn ship to upright position• Upsetting moment:Upsetting moment: tends to tends to

overturn shipoverturn ship• Magnitude of righting Magnitude of righting

moment:moment:• RM = W * GZRM = W * GZ (ft-tons) (ft-tons)• GZ: moment armGZ: moment arm (ft) (ft)

MetacenterMetacenter

• Def’n: the intersection Def’n: the intersection of two successive of two successive lines of action of the lines of action of the force of buoyancy as force of buoyancy as ship heels through ship heels through small angles (M)small angles (M)• If angle too large, M If angle too large, M

moves off centerlinemoves off centerline

MetacenterMetacenter• Metacentric Height Metacentric Height

(GM)(GM)• Determines size of Determines size of

righting/upsetting arm righting/upsetting arm (for angles < 7(for angles < 7oo))

GZ = GM*sinGZ = GM*sinφφ• Large GM -> large Large GM -> large

righting arm (stiff)righting arm (stiff)• Small GM -> small Small GM -> small

righting arm (tender)righting arm (tender)

MetacenterMetacenter• Relationship between G and MRelationship between G and M

• G under M: ship is stableG under M: ship is stable• G = M: ship neutralG = M: ship neutral• G over M: ship unstableG over M: ship unstable

STABLESTABLE UNSTABLEUNSTABLE

Metacenter v. Stability CurvesMetacenter v. Stability Curves• At this point, we could use lots of At this point, we could use lots of

trigonometry to determine exact values trigonometry to determine exact values of forces, etc for all angles -> too much of forces, etc for all angles -> too much work work

• GM used as a measure of stability up to GM used as a measure of stability up to 7°, after that values of GZ are plotted at 7°, after that values of GZ are plotted at successive angles to create the stability successive angles to create the stability curvecurve

Stability CurveStability Curve

Stability CurveStability Curve• Plot GZ (righting arm) vs. angle of heelPlot GZ (righting arm) vs. angle of heel

• Ship’s G does not change as angle changesShip’s G does not change as angle changes• Ship’s B always at center of underwater portion of Ship’s B always at center of underwater portion of

hullhull• Ship’s underwater portion of hull changes as heel Ship’s underwater portion of hull changes as heel

angle changesangle changes• GZ changes as angle changesGZ changes as angle changes

Questions?Questions?