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HULL FORM AND GEOMETRY Intro to Ships and Naval Engineering (2.1)

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HULL FORM AND GEOMETRY Intro to Ships and Naval Engineering (2.1) Factors which influence design: –Size – Speed – Seakeeping – Maneuverability – Stability – Special Capabilities (Amphib, Aviation,...) Compromise is required!

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HULL FORM AND GEOMETRY Categorizing Ships (2.2) Methods of Classification: – 1.0 Usage: Merchant Ships (Cargo, Fishing, Drill, etc) Naval and Coast Guard Vessels Recreational Boats and Pleasure Ships Utility Tugs Research and Environmental Ships Ferries

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HULL FORM AND GEOMETRY Categorizing Ships (2.2) Methods of Classification (con’t): – 2.0 Physical Support: Hydrostatic Hydrodynamic Aerostatic (Aerodynamic)

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HULL FORM AND GEOMETRY Categorizing Ships (2.2)

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HULL FORM AND GEOMETRY Categorizing Ships (2.2) Hydrostatic Support (also know as Displacement Ships) Float by displacing their own weight in water – Includes nearly all traditional military and cargo ships and 99% of ships in this course – Small Waterplane Area Twin Hull ships (SWATH) – Submarines

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HULL FORM AND GEOMETRY Categorizing Ships (2.2) Aerostatic Support - Vessel rides on a cushion of air. Lighter weight, higher speeds, smaller load capacity. – Air Cushion Vehicles - LCAC: Opens up 75% of littoral coastlines, versus about 12% for displacement – Surface Effect Ships - SES: Fast, directionally stable, but not amphibious

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HULL FORM AND GEOMETRY Categorizing Ships (2.2) Hydrodynamic Support - Supported by moving water. At slower speeds, they are hydrostatically supported – Planing Vessels - Hydrodynamics pressure developed on the hull at high speeds to support the vessel. Limited loads, high power requirements. – Hydrofoils - Supported by underwater foils, like wings on an aircraft. Dangerous in heavy seas. No longer used by USN. (USNA Project!)

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HULL FORM AND GEOMETRY Categorizing Ships (2.2) Hydrostatic Support - Based on Archimedes Principle – 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.”

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HULL FORM AND GEOMETRY Categorizing Ships (2.2) Archimedes Principle - The Equation where: F B = is the magnitude of the resultant buoyant force in lb = (“rho”) density of the fluid in lb s 2 / ft 4 or slug/ft 3 g = magnitude of accel. due to gravity (32.17 ft/s 2 ) = volume of fluid displaced by the object in ft 3

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HULL FORM AND GEOMETRY How are these vessels supported? Hydrostatic Hydrodynamic Aerostatic A combination?

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HULL FORM AND GEOMETRY

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Brain Teasers!

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HULL FORM AND GEOMETRY

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Representing Ship Designs Problems include: – Terms to use (jargon) – How to represent a 3-D object on 2-D paper Sketches Drawings Artist’s Rendition

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HULL FORM AND GEOMETRY Basic Dimensions (2.3.3) Design Waterline (DWL) - The waterline where the ship is designed to float. Stations - Parallel planes from forward to aft, evenly spaced (like bread). Normally an odd number to ensure an even number of blocks.

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HULL FORM AND GEOMETRY Basic Dimensions (2.3.3) Forward Perpendicular (FP) - Forward station where the bow intersects the DWL. Station 0. Aft Perpendicular (AP) - After station located at either the rudder stock or the intersection of the stern with the DWL. Station 10. Length Between Perpendiculars (Lpp) -Distance between the AP and the FP. In general the same as LWL (length at waterline).

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HULL FORM AND GEOMETRY Basic Dimensions (2.3.3) Length Overall (LOA) - Overall length of the vessel. Midships Station ( ) - Station midway between the FP and the AP. Station 5 in a 10-station ship. Also called amidships.

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HULL FORM AND GEOMETRY Hull Form Representation ( ) Lines Drawings - Traditional graphical representation of the ship’s hull form. “Lines” Half-Breadth Sheer Plan Body Plan

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HULL FORM AND GEOMETRY Hull Form Representation ( ) Lines Plan Half- Breadth Plan Sheer Plan Body Plan

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HULL FORM AND GEOMETRY Hull Form Representation ( ) Half-Breadth Plan (“Breadth” = “Beam”)

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HULL FORM AND GEOMETRY Hull Form Representation ( ) Half-Breadth Plan (“Breadth” = “Beam’) – Intersection of horizontal planes with the hull to create waterlines. (Parallel with water.)

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HULL FORM AND GEOMETRY Hull Form Representation ( ) Sheer Plan – Parallel to centerplane – Pattern for construction of longitudinal framing.

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HULL FORM AND GEOMETRY Hull Form Representation ( ) Sheer Plan – Intersection of planes parallel to the centerline plane define the Buttock Lines. These show the ship’s hull shape at a given distance from the centerline plane.

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HULL FORM AND GEOMETRY Hull Form Representation ( ) Body Plan – Pattern for construction of transverse framing.

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HULL FORM AND GEOMETRY Hull Form Representation ( ) Body Plan – Intersection of planes parallel to the centerline plane define the Section Lines. – Section lines show the shape of the hull from the front view for a longitudinal position

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HULL FORM AND GEOMETRY Table of Offsets (2.4) The distances from the centerplane are called the offsets or half-breadth distances.

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HULL FORM AND GEOMETRY Table of Offsets (2.4) Used to convert graphical information to a numerical representation of a three dimensional body. Lists the distance from the center plane to the outline of the hull at each station and waterline. There is enough information in the Table of Offsets to produce all three lines plans.

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HULL FORM AND GEOMETRY Hull Form Characteristics (2.5) Depth (D) - Distance from the keel to the deck. Remember “Depth of Hold.” Draft (T) - Distance from the keel to the surface of the water. Beam (B) - Transverse distance across each section. Half-Breadths are “half of beam”. Flare Tumblehome

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HULL FORM AND GEOMETRY Hull Form Characteristics (2.5) Keel (K) - Reference point on the bottom of the ship and is synonymous with the baseline.

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HULL FORM AND GEOMETRY Centroids (2.6) Centroid – The geometric center of a body. Center of Mass - A “single point” location of the mass. – Better known as the Center of Gravity (CG). CG and Centroids are only in the same place for uniform (homogenous) mass!

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HULL FORM AND GEOMETRY Centroids (2.6) Centroids and Center of Mass can be found by using a weighted average.

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HULL FORM AND GEOMETRY What is the longitudinal center of gravity of this 18 foot row boat? Hull: 150 lb at station 6 Seat: 10 lb at station 5 Rower: 200 lb at station 5.5

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HULL FORM AND GEOMETRY Center of Flotation (F or CF) (2.7.1) The centroid of the operating waterplane. (The center of an area.) The point about which the ship will list and trim! Transverse Center of Flotation (TCF) - Distance of the Center of Flotation from the centerline.(Often = 0 feet)

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HULL FORM AND GEOMETRY Center of Flotation (F or CF) (2.7.1) Longitudinal Center of Flotation (LCF) - Distance from midships (or the FP or AP) to the Center of Flotation. The Center of Flotation changes as the ship lists or trims because the shape of the waterplane changes.

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HULL FORM AND GEOMETRY Center of Buoyancy (B or CB) (2.7.2) Centroid of the Underwater Volume. Location where the resultant force of buoyancy (F B ) acts. Transverse Center of Buoyancy (TCB) - Distance from the centerline to the Center of Buoyancy.

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HULL FORM AND GEOMETRY Center of Buoyancy (B or CB) (2.7.2) Vertical Center of Buoyancy (VCB or KB) - Distance from the keel to the Center of Buoyancy. Longitudinal Center of Buoyancy (LCB) - Distance from the amidships or AP or FP to the Center of Buoyancy. Center of Buoyancy moves when the ship lists or trims (TCB).

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HULL FORM AND GEOMETRY Center of Buoyancy (B or CB) (2.7.2) Which way is it moving? Fwd or Aft?

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HULL FORM AND GEOMETRY Fundamental Geometric Calculations (2.8) A ship’s hull is a complex shape which cannot be described by a mathematical equation! How can centroids, volumes, and areas be calculated? (Hint: you can’t integrate!) Use Numerical Methods to approximate an integral! – Trapezoidal Rule (linear approximation) – Simpson’s Rule (quadratic approximation)

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HULL FORM AND GEOMETRY Fundamental Geometric Calculations (2.8.1) Example: Waterplane Calculation (Trapezoidal)

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HULL FORM AND GEOMETRY Fundamental Geometric Calculations (2.8.1) Simpson’s Rule - Used to integrate a curve with an odd number of evenly spaced ordinates. (Ex. Stations )

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Area under the curve between -s and s: Solving this equation for the given endpoints: A simple example with a rectangle... HULL FORM AND GEOMETRY Fundamental Geometric Calculations (2.8.1)

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HULL FORM AND GEOMETRY Fundamental Geometric Calculations (2.8.1) If the curve extends over more than three points the equation becomes: “s” is the spacing between ordinates. Usually will be the spacing between stations or waterlines.

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HULL FORM AND GEOMETRY Section (2.9) Using Simpson’s 1st Rule, you must* be able to calculate: – Waterplane Area – Sectional Area – Submerged Volume – Longitudinal Center of Flotation (LCF) * meaning: “this will be on the homework, labs, quizzes, and exams!”

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Applying Simpson’s Rule (2.9) Methodology – Draw a picture of what you intend to integrate. – Show the differential element you are using. – Properly label your axis and drawing. – Write out the generalized calculus equation in the proper symbols (optional). HULL FORM AND GEOMETRY

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Applying Simpson’s Rule (2.9) Methodology (con’t) – Write out Simpson’s Equation in generalized form (if a curved shape). – Substitute each number in the generalized Simpson’s Equation. – Calculate the final answer. HULL FORM AND GEOMETRY

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Waterplane Area (2.9.1) Numerically integrate the half-breadth as a function of the length of the vessel. HULL FORM AND GEOMETRY

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Waterplane Area (2.9.1) Writing out the Simpson’s equation: where: A wp is the waterplane area in ft 2 s is the Simpson’s spacing y(x) is the “y” offset or half-breadth at each value of “x” in ft Example for a ship! HULL FORM AND GEOMETRY

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Section Area (2.9.2) Numerical integration of the half- breadth as a function of the draft. HULL FORM AND GEOMETRY

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Section Area (2.9.2) Determine how to find the area(s) by using which methods (Simpson’s must be an odd number of points!) Writing out the generalized Simpson’s Equation and the triangle equation: HULL FORM AND GEOMETRY

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Archimedes Principle - The Equation Recall that the goal of us using the Lines Plan And the Table of Offsets was to find the Volume…, and hence the buoyant force! And, if in static equilibrium, then F B =Weight! But so far, we can only calculate the section and waterplane areas…

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Submerged Volume: Longitudinal Integration (2.9.3) Integration of the section areas over the length of the ship. “Curve of Areas” HULL FORM AND GEOMETRY What is a barge’s section area, volume and curve of areas if is 100 ft long, 25 feet beam and 10 feet draft? “Curve of Areas” Stn4

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HULL FORM AND GEOMETRY What is a barge’s section area, volume, curve of areas and displacement? Section Area = Beam x Draft Volume = Section Area x Length 100 ft long, 25 feet beam and 10 feet draft FPAP Curve of Areas Section Area

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Submerged Volume: Longitudinal Integration (2.9.3) HULL FORM AND GEOMETRY So, the volume if using Simpson’s is: Ques: where is the “2”?

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Longitudinal Center of Flotation (LCF) (2.9.4) (Centroid of Waterplane Area) HULL FORM AND GEOMETRY Point at which the vessel “___” and “___”? Distance from the Forward Perpendicular to the center of flotation (or from MP). Found as a weighted average of the distance from the Forward Perpendicular multiplied by the ratio of the half-breadth to the total waterplane area.

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Longitudinal Center of Flotation (LCF) (2.9.4) (Centroid of Waterplane Area) HULL FORM AND GEOMETRY Drawing of the LCF: Recall: For most “normal” vessels LCF is between Stn 5 and 6.7

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Longitudinal Center of Flotation (LCF) (2.9.4) (Centroid of Waterplane Area) HULL FORM AND GEOMETRY Writing the general calculus equation and the general Simpson’s form (for 4 Simpson’s spaces in a 10 station ship):

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Sample Quiz Questions! The Center of Flotation is: a.Centroid of the underwater volume b.Point at which Fb acts c.Centroid of the waterplane d.Point at which the hydrostatic force acts To calculate the submerged volume of a ship, one would a. Integrate half-breadths from the keel to the waterplane b. Integrate half-breadths longitudinally at the waterline c. Integrate section areas longitudinally d. Use Simpson’s Rule to integrate waterplane areas at each station

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Curves of Form (2.10) HULL FORM AND GEOMETRY WHAT THEY ARE: Graphical representation of the ship’s geometric-based properties. WHY: When weight is added, removed or shifted, the underwater shape changes and therefore the geometric properties change. DETAILS: Based on a given average draft. Unique for every vessel. The ship is assumed to be in seawater.

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Curves of Form (2.10) HULL FORM AND GEOMETRY Curves of Form Include: – Displacement – LCB – VCB – Immersion (TPI) – LCF – MT1 – And some others...

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Curves of Form (2.10) HULL FORM AND GEOMETRY

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Curves of Form ( ) HULL FORM AND GEOMETRY Longitudinal Center of Buoyancy (LCB): – The distance in feet from the longitudinal reference position to the center of buoyancy. – The reference position could be the FP or midships. If it is midships remember that distances aft of midships are negative!

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Curves of Form ( ) HULL FORM AND GEOMETRY Vertical Center of Buoyancy (VCB): – The distance in feet from the baseplane to the center of buoyancy. – Sometimes this distance is labeled KB with a bar over the letters.

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Curves of Form ( ) HULL FORM AND GEOMETRY Tons Per Inch Immersion (TPI): – TPI is defined as the tons required to obtain one inch of sinkage in salt water. – Parallel sinkage is when the ship changes its forward and after drafts by the same amount so that no change in trim occurs.

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Curves of Form ( ) HULL FORM AND GEOMETRY An approximate formula for TPI based on the area of the waterplane can be derived as follows:

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Curves of Form ( ) HULL FORM AND GEOMETRY Longitudinal Center of Flotation (LCF): The distance in feet from the longitudinal reference point to the center of flotation. The reference position could be the FP or midships. If it is midships remember that distances aft of midships are negative.

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Curves of Form ( ) HULL FORM AND GEOMETRY Moment to Trim One Inch (Moment/ Trim 1” or MT1"): – The ship will rotate about the (?) when a moment is applied to it. – The moment can be produced by adding, removing, or shifting a weight some distance from the center of flotation. – The units are?

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Curves of Form ( ) HULL FORM AND GEOMETRY Trim is defined as the change in draft aft minus the change in draft forward. – If the ship starts level and trims so that the forward draft increases by 2 inches and the aft draft decreases by 1 inch, the trim would be -3 inches.

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Curves of Form ( ) HULL FORM AND GEOMETRY Since a ship is typically wider at the stern than at the bow, the center of flotation will typically be aft of midships. – This means that when a ships trims, it will typically have a greater change in the forward draft than in the after draft.

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Curves of Form ( ) HULL FORM AND GEOMETRY KM L : (A measure of pitch stability) – The distance in feet from the keel to the longitudinal metacenter. – This distance is on the order of one hundred to one thousand feet whereas the distance from the keel to the transverse metacenter is only on the order of ten to thirty feet.

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Curves of Form ( ) HULL FORM AND GEOMETRY KM T : (A measure of roll stability) – This is the distance in feet from the keel to the transverse metacenter. – Typically, we do not bother putting the subscript “T” for any property in the transverse direction because it is assumed that when no subscript is present the transverse direction is implied.

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The End of Chapter 2 Did you meet all the chapter’s objectives?! In one word… buoyancy!

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