1. Stability & Buoyancy

2. Objectives Principles of Stability Archimedes Principle
Terminology of ship’s hydrostatics
Stability & moments -> staying upright
Metacenter, Center of Gravity, Center of Buoyancy, etc.
Stability curves

3. Principles of Stability
Floating object is acted on by forces of gravity and forces of buoyancy
Static equilibrium SFi = 0
Three conditions of static equilibrium:
Stable: return to same position if tipped
Neutral: when rotated, will come to rest in any position
Unstable: will come to rest in new position if force acts on it

4. Archimedes Principle
Law: a body floating or submerged in a fluid is buoyed up by a force equal to the weight of the water it displaces
Depth to which ship sinks depends on density of water (r = 1 ton/35ft3 seawater)

5. Archimedes Principle
Ship sinks until weight of water displaced by the underwater volume is equal to the weight of the ship
Forces of gravity: G = mshipg =Wship
Forces of buoyancy: B = rwaterVdisplaced
Wship = rwaterVdisplaced

6. Archimedes Principle
Forces act everywhere on ship -> too tough to analyze
Center of Gravity (G): all gravity forces as one force acting downward through ship’s geometric center
Center of Buoyancy (B): all buoyancy forces as one force acting upward through underwater geometric center

7. Archimedes Principle Center of Gravity (G): Center of Buoyancy (B):
Changes position only by change/shift in mass of ship
Does not change position with movement of ship
Center of Buoyancy (B):
Changes position with movement of ship -> underwater geometric center moves
Also affected by displacement G

8. Hydrostatics Terminology
Displacement: total weight of ship = total submerged volume of ship (measured in tons)
Draft: vertical distance from waterline to keel at deepest point (measured in feet)
Reserve Buoyancy: volume of watertight portion of ship above waterline (important factor in ship’s ability to survive flooding)
Freeboard: vertical distance from waterline to main deck (rough indication of reserve buoyancy)

9. Hydrostatics Terminology
As draft & displacement increase, freeboard and reserve buoyancy decrease

10. Moments
Def’n: tendency of a force to produce rotation or to move an object about an axis
Distance between the force and axis of rotation is the moment arm
Couple: two forces of equal magnitude in opposite and parallel directions, separated by a perpendicular distance
G and B are a couple

11. Moments Depending on location of G and B, two types of moments:
Righting moment: tends to return ship to upright position
Upsetting moment: tends to overturn ship
Magnitude of righting moment:
RM = W * GZ (ft-tons)
GZ: moment arm (ft)

12. Metacenter
Def’n: the intersection of two successive lines of action of the force of buoyancy as ship heels through small angles (M)
If angle too large, M moves off centerline

13. Metacenter Metacentric Height (GM)
Determines size of righting/upsetting arm (for angles < 7o)
GZ = GM*sinf
Large GM -> large righting arm (stiff)
Small GM -> small righting arm (tender)

14. Metacenter Relationship between G and M G under M: ship is stable
G = M: ship neutral
G over M: ship unstable
STABLE
UNSTABLE

15. Metacenter v. Stability Curves
At this point, we could use lots of trigonometry to determine exact values of forces, etc for all angles -> too much work
GM used as a measure of stability up to 7°, after that values of GZ are plotted at successive angles to create the stability curve

16. Stability Curve

17. Stability Curve Plot GZ (righting arm) vs. angle of heel
Ship’s G does not change as angle changes
Ship’s B always at center of underwater portion of hull
Ship’s underwater portion of hull changes as heel angle changes
GZ changes as angle changes

18. Questions?