Overview Chapter 3 - Buoyancy versus gravity = stability (see Chapter Objectives in text) Builds on Chapters 1 and 2 6-week exam is Chapters 1-3!

HYDROSTATICS Review (3.1) Archimedes Principle: “An object partially or fully submerged in a fluid will experience a resultant vertical force equal in magnitude to the weight of the volume of fluid displaced by the object.” This force is called the “buoyant force” or the “force of buoyancy”(FB).

HYDROSTATICS Review (3.1) Mathematical Equation: Where. . . FB is the magnitude of the resultant buoyant force in lb,  is the density of the fluid in lb s2 / ft4 , g is the magnitude of the acceleration of gravity normally taken to be 32.17 ft / s2 .  is the volume of fluid displaced by the object in ft3 .

Hydrostatics Vessel Degrees of Freedom And Static Equilibrium The forces lead to translations: Heave Surge Sway The moments lead to rotations: Roll Pitch Yaw

Static Equilibrium : Forces and Moments (3.1.2.1-2) HYDROSTATICS Static Equilibrium : Forces and Moments (3.1.2.1-2) Sum of the Resultant Forces: Sum of the Moments about a reference point: Static equilibrium must consist of both conditions!

Static Equilibrium : Stability HYDROSTATICS Static Equilibrium : Stability t B Is this boat in static equilibrium? What are the component forces and moments? Are they internal or external?

HYDROSTATICS Static Equilibrium (3.1.1.2) Wave? What is the hydrostatic pressure? F=p*A

Static Equilibrium : Stability (3.2) HYDROSTATICS Static Equilibrium : Stability (3.2) t B

Changes in the Center of Gravity (3.2) HYDROSTATICS Changes in the Center of Gravity (3.2) The Center of Gravity (G) is the point at which all of the mass of the ship can be considered to be located (for most problems). Terminology! UPPERCASE for ship; lowercase for a smaller weight. It is referenced vertically from the keel of the ship (KG or VCG or Kg). (1) Shifting, (2) adding, or (3) removing weight changes the location of the Center of Gravity.

Static Equilibrium : Stability HYDROSTATICS Static Equilibrium : Stability t B Where is the Center of Gravity? The Center of Buoyancy? Are they vertically aligned? Why/Why not?

Changes in the Center of Gravity (3.2.1.1) HYDROSTATICS Changes in the Center of Gravity (3.2.1.1) When weight is added to a ship, the CG will move in a straight line from its current position toward the cg of the weight being added. G0 to Gf. The distance is a ratio of the weight and disp. What happens to the Center of Buoyancy (and the ship)?

Changes in the Center of Gravity (3.2.1.2) HYDROSTATICS Changes in the Center of Gravity (3.2.1.2) When weight is removed from a ship, G will move in a straight line from its current position away from the center of gravity of the weight being removed. G0 to Gf.

Changes in the Center of Gravity (3.2.1.3) HYDROSTATICS Changes in the Center of Gravity (3.2.1.3) When a small weight is shifted (but not added or removed, CG will move parallel to the weight shift but a much smaller distance because it is only a small fraction of the total weight of the ship.

Vertical Shift in the Center of Gravity (3.2.2.1) HYDROSTATICS Vertical Shift in the Center of Gravity (3.2.2.1) Where: (note: some use the term “initial” for “old” and “final” for “new” KGnew is the final vertical position of the center of gravity of the ship as referenced from the keel. KG’s are in “feet”. KGold is the initial vertical position of the center of gravity of the ship as referenced from the keel.

Vertical Shift in the Center of Gravity (3.2.2.1) HYDROSTATICS Vertical Shift in the Center of Gravity (3.2.2.1) And, s new is the final displacement of the ship in LT. In this example, it is equal to the initial displacement plus or minus the weight added. s old is the initial displacement of the ship in LT. Kg added weight is the vertical position of the center of gravity of the weight being added as referenced from the keel. This line segment is a distance in feet. w added weight is the weight of the weight to be added in LT.

Vertical Shift in the Center of Gravity (3.2.2.1) HYDROSTATICS Vertical Shift in the Center of Gravity (3.2.2.1) The first equation was for a weight addition or removal. What do we do for a weight shift? What is different Re-examine our first vertical shift equation. What changes?

Vertical Shift in the Center of Gravity (3.2.2.3) HYDROSTATICS Vertical Shift in the Center of Gravity (3.2.2.3) So, the final equation for vertical shifts is: Example: A 150 pound person climbs in a 10 pound canoe and sits down. How much has KG shifted? KGold=0.5 ft Kg=?

Vertical Shift in the Center of Gravity (3.2.2.4) HYDROSTATICS Vertical Shift in the Center of Gravity (3.2.2.4) Last Comments: The general equation covers all cases for a change in KG. This is the equation you should apply to the exams!

Transverse Shift in the Center of Gravity (3.2.3) HYDROSTATICS Transverse Shift in the Center of Gravity (3.2.3) Shifts “side to side” of the Center of Gravity. Starboard is positive and port is negative! As in Vertical case, the Transverse movement of “G” may be caused by either (1) addition, (2) removal, or (3) shifting of weights.

Transverse Shift in Center of Gravity (3.2.3) HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3) Results in a “List” on the Vessel. “List” occurs when a vessel is in static equilibrium and down by either the port or starboard side. No external forces are required to maintain this condition and it is permanent unless the Center of Gravity changes. “List” is different from “heeling”. Heeling occurs because an external couple is acting on the vessel. Heeling is a more temporary condition.

Transverse Shift in Center of Gravity (3.2.3) HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3) Example (Listing or Heeling?)

Transverse Shift in Center of Gravity (3.2.3) HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3)

Transverse Shift in Center of Gravity (3.2.3) HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3) The Transverse Center of Gravity is referenced in the transverse (athwartships) direction from the centerline of the ship and is labeled TCG. The equation used for a transverse shift in the Center of Gravity is the same as was used for the vertical shift! (With some changes in the notation.)

Transverse Shift in Center of Gravity (3.2.3.4) HYDROSTATICS Transverse Shift in Center of Gravity (3.2.3.4) Remember a weight shift is just like removing a weight from its original location and adding it to its final location. So for just a weight shift, the generalized equation simplifies to: Example: Your 100 LT ship is initially upright. You pump 5 LT of water from a point 15 ft starboard of centerline to 10 ft port of centerline. What is the new TCG? (We will use that answer later to find the angle of heel.”

Vertical and Transverse Changes in “G” The Key Equations! HYDROSTATICS Vertical and Transverse Changes in “G” The Key Equations! When faced with a change in weight (add, sub or move), first sketch it, then solve KG, then solve TCG!

HYDROSTATICS Metacenter (3.3) A reference point for hydrostatic calculations for small angles of roll (less than 10 degrees) or pitch (less than five degrees). Defined as the intersection of the buoyancy forces and the ship centerline.

HYDROSTATICS Metacenter (3.3) The higher the metacenter, the more stable the ship is! There is a different metacenter for ship pitching in the longitudinal direction and ships rolling in the transverse direction. BMT is for roll, BML for pitch. Which is higher? If the subscript is omitted, it means BMT.

HYDROSTATICS Metacentric Radius (3.3.1.1) The distance from the Metacenter to the Center of Buoyancy is defined as the Metacentric Radius (BM). MT B K

Quick Review Finding KMT from the Curves of Form For a draft of 10 ft… Gen’l Scale = 192 192*0.06 ft KMT=11.5 ft

HYDROSTATICS Metacentric Height (3.3.1.2) MT G The distance between the Center of Gravity (G) and the Metacenter (M) is defined as the Metacentric Height (GM). MT G

HYDROSTATICS Metacentric Height (3.3.1.2) Why is GM important? If G is below M, then GM is said to be positive. The ship does not want to capsize. This is GOOD! If G coincides with M, then GM is said to be zero. A vessel would stay heeled. This is not very good. If G is above M, the GM is said to be negative. The ship will tip over. This is REALLY BAD!

Good (positive) GM! The ship wants to roll upright. B

Bad (negative) GM. The ship wants to roll over Bad (negative) GM! The ship wants to roll over. G is either too high or M is too low! G MT B

HYDROSTATICS Metacentric Radius (3.3.2.2) B and M are functions of the hull shape and are generally constant over the life of the ship. G is based on the weights and changes constantly. To be safe at sea, we need to find the ship’s KG to make sure it is sufficiently below M! KG = KM - GM Where, KM is shown on the Curves of Form. GM is found from both calculations and by an Inclining Experiement

HYDROSTATICS Metacentric Radius (3.3.2.2) KM=KB + BM where: KB is found by numerical integration but for most vessels is between 40-50% of the draft BM is found by: IT is the “Transverse Moment of Area of the Waterplane” and has the units of ft4 For a box-shaped barge it simplifies to:

Example stability check You have just bought a 30-foot long floating dock, made some modifications and will now put it in the water. It is 6 ft wide and 2.5 feet deep. KG=2 ft and it has a 1 ft draft. Will it be stable? (eg Find GM and determine if it is positive!)

Calculating Angle of List (3.4) HYDROSTATICS Calculating Angle of List (3.4) As a weight shifts across the deck of a vessel, the vessel lists (or “inclines”. How can we predict the angle of inclination (list)? Derivation of Equation Draw two vessels, one upright and one listing. Show a weight moving, along with the CG and B.

Calculating Angle of List (3.4)

Calculating Angle of List (3.4.2) HYDROSTATICS Calculating Angle of List (3.4.2) The weight is shifted causing a shift in the Center of Gravity. A moment is created causing the vessel to incline. The underwater shape of the hull changes causing the Center of Buoyancy (B) to move until it is in line with the Center of Gravity (G) and the vessel is back in static equilibrium.

Calculating Angle of List (3.4.3) HYDROSTATICS Calculating Angle of List (3.4.3) From the geometry and then some substitution, we get: M W f t G B

Calculating Angle of List (3.4.3) HYDROSTATICS Calculating Angle of List (3.4.3) This equation only works for small angles because it assumes that the Metacenter does not move! Note that for small angles, tan = sin! So you can calculate GM from either along the old or new inclined axis. Example: You move a 1 LT weight 25 feet to starboard on your 100 LT ship and it lists 2 degrees. What is GM? How would you find KG?

Inclining Experiment (3.5) HYDROSTATICS Inclining Experiment (3.5) Uses small-angle hydrostatics to find the vertical center of gravity (KG) of a ship. Process: A weight is moved a transverse distance, causing a shift in the TCG, and resulting in measurable inclination (list).

Inclining Experiment (3.5) HYDROSTATICS Inclining Experiment (3.5) Navy 44 Incline Experiment

Inclining Experiment (3.5.1) HYDROSTATICS Inclining Experiment (3.5.1) Solving the Angle of Heel equation for the metacentric height (GM), we find: The easiest way to do this experiment is to use one set of weights at one distance off centerline. Alas, this would have significant experimental errors, so we measure the inclination with different weights and different positions.

Inclining Experiment (3.5.1) HYDROSTATICS Inclining Experiment (3.5.1) We then plot the data on a graph where the y- axis is the Inclining Moment (wt) and the x- axis is the Tangent of the inclining angle (Tan ). The average value of GM can be found from the slope of the line. We can see that:

Inclining Experiment (3.5.1) HYDROSTATICS Inclining Experiment (3.5.1) Recall: We want to find the Center of Gravity which can be found by the equation: KG=KM-GM KM is found from the Curves of Form GM is found from the Inclining Experiment

Inclining Experiment (3.5.2) HYDROSTATICS Inclining Experiment (3.5.2) Removing the Inclining Apparatus we must recalculate KG. This is done as a weight removal problem:

Inclining Experiment (3.5.3) HYDROSTATICS Inclining Experiment (3.5.3) Shipboard Considerations: No initial list. Minimum trim. Dry bilges. Liquid fuel and oil to be in accordance with the Shipyard Memo. Sluice valves closed. All consumables are to be inventoried. Minimum number of personnel remain onboard. See the example in your text!

Longitudinal Changes in the Center of Gravity (3.6) HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) A longitudinal shift in the CG will result in the vessel having some trim. Trim is the difference between the forward and aft drafts, Tf and Ta. It may be calculated by: Ex. A ship has a draft of 15’ fwd and 16’ aft. Trim = 1 ft The Mean Draft is:

Longitudinal Changes in the Center of Gravity (3.6) HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) A vessel is trimmed by the bow when the bow has a deeper draft. This is indicated by a negative trim. A vessel is trimmed by the stern when the stern has a deeper draft. This is indicated by a positive trim. What is the point which the vessel trims about?

Longitudinal Changes in the Center of Gravity (3.6) HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) What happens when a weight is shifted forward or aft? The vessel goes down by the bow or stern depending on the direction of the weight shift. Note that the change in trim is independent of the original location of the weight. (i.e. It only matters whether the weight moves forward or aft)

Longitudinal Changes in the Center of Gravity “The Trim Problem” (3.6) HYDROSTATICS Longitudinal Changes in the Center of Gravity “The Trim Problem” (3.6) Draw a picture of what is happening when a vessel trims due to a weight shift:

Longitudinal Changes in the Center of Gravity (3.6) HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) As the weight shifts forward, a new operating waterline is created and the draft decreases aft and increases forward. dTaft dTfwd

Longitudinal Changes in the Center of Gravity (3.6) HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) We now have two similar triangles and will draw a third which represents the change in trim. Recall: Trim = Taft -Tfwd So the total Trim with a change in trim is: And with no initial trim, then the change in TRIM is:

Longitudinal Changes in the Center of Gravity (3.6) HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) To calculate the final drafts we will need to find: Where MT1 is from the Curves of Form (2.10) We use similar triangles (ratios) to find the change in draft due to the weight shift.

Longitudinal Changes in the Center of Gravity (3.6) HYDROSTATICS Longitudinal Changes in the Center of Gravity (3.6) Example: You have a 1000 x 200 x 90 foot tanker (100,000 LT) with F at Stn 6. It has zero TRIM. You move 1000 LT of oil 450 ft aft. What is the new draft at the stern? MT1~ IL/420L=150,000 FTLT/in