chapter 2

HULL FORM AND GEOMETRYIntro to Ships and Naval Engineering (2.1)

HULL FORM AND GEOMETRYIntro to Ships and Naval Engineering (2.1)

• Factors which influence design:–Size– Speed– Seakeeping– Maneuverability– Stability– Special Capabilities (Amphib, Aviation, ...)

Compromise is required!

HULL FORM AND GEOMETRYCategorizing 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


Methods of Classification (con’t):

– 2.0 Physical Support:

Hydrostatic Hydrodynamic Aerostatic (Aerodynamic)


• 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


• 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


• 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!)


• 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.”


• Archimedes Principle - The Equation

gFB where: FB = is the magnitude of the resultant buoyant force in lb

= (“rho”) density of the fluid in lb • s2 / ft 4 or slug/ft3

g = magnitude of accel. due to gravity (32.17 ft/s2)

= volume of fluid displaced by the object in ft3

HULL FORM AND GEOMETRY

How are these vessels supported?

• Hydrostatic

• Hydrodynamic

• Aerostatic

• A combination?


Brain Teasers!

HULL FORM AND GEOMETRYRepresenting Ship Designs

• Problems include:

– Terms to use (jargon)

– How to represent a 3-D object on 2-D paper Sketches Drawings Artist’s Rendition

HULL FORM AND GEOMETRYBasic 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.


• 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).


• 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.

HULL FORM AND GEOMETRYHull Form Representation (2.3.0-2.3.3)

• Lines Drawings - Traditional graphical representation of the ship’s hull form. “Lines”

Half-Breadth

Sheer Plan

Body Plan


Lines Plan

Half-Breadth Plan

Sheer Plan

Body Plan


• Half-Breadth Plan (“Breadth” = “Beam”)


• Half-Breadth Plan (“Breadth” = “Beam’)

– Intersection of horizontal planes with the hull to create waterlines. (Parallel with water.)


• Sheer Plan– Parallel to centerplane– Pattern for construction of longitudinal framing.


• 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.


• Body Plan– Pattern for construction of transverse framing.


• 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

HULL FORM AND GEOMETRYTable of Offsets (2.4)

• The distances from the centerplane are called the offsets or half-breadth distances.

HULL FORM AND GEOMETRYTable 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.

HULL FORM AND GEOMETRYHull 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

HULL FORM AND GEOMETRYHull Form Characteristics (2.5)

• Keel (K) - Reference point on the bottom of the ship and is synonymous with the baseline.

Beam (B)

Depth (D)

CL

Draft (T)

Freeboard

Camber

Typical view of the midship section of a ship.

K•

W L

HULL FORM AND GEOMETRYCentroids (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!

HULL FORM AND GEOMETRYCentroids (2.6)

a1a2

a3an

y1y2

y3 yn

Y

X

• Centroids and Center of Mass can be found by using a weighted average.

1i i

1i iiave

a

ayy

321

332211ave aaa

ayayayy


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

20010150

5.52005106150LCG

bowthefromft25.10or

7.5StnLCG

HULL FORM AND GEOMETRYCenter 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 The point about which the ship will list and trimwill list and trim!!

• Transverse Center of Flotation (TCF) - Distance of the Center of Flotation from the centerline.(Often = 0 feet)

HULL FORM AND GEOMETRYCenter 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.

HULL FORM AND GEOMETRYCenter of Buoyancy (B or CB) (2.7.2)

• Centroid of the Underwater Volume.

• Location where the resultant force of buoyancy (FB) acts.

• Transverse Center of Buoyancy (TCB) - Distance from the centerline to the Center of Buoyancy.


• 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).


Which way is it moving? Fwd or Aft?

HULL FORM AND GEOMETRYFundamental 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)

HULL FORM AND GEOMETRYFundamental Geometric Calculations (2.8.1)

Example: Waterplane Calculation (Trapezoidal)


Fundamental Geometric Calculations (2.8.1)

• Simpson’s Rule - Used to integrate a curve with an odd number of evenly spaced ordinates. (Ex. Stations 0 - 10)

P (-s, y )o o

P (0, y )1 1

P (s, y )2 2

y

x-s s0

y(x) cx2 dx e2( )y x cx dx e

• Area under the curve between -s and s:

• Solving this equation for the given endpoints:

• A simple example with a rectangle...


Fundamental Geometric Calculations (2.8.1)

2 2( ) (2 6 )3

s

s

sArea cx dx e dx cs e

0 1 2( 4 )3

sArea y y y

HULL FORM AND GEOMETRYFundamental 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.

0 1 2 3( 4 2 4 ... )3 n

sArea y y y y y

HULL FORM AND GEOMETRYSection (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!”

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).


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.


Waterplane Area (2.9.1)

• Numerically integrate the half-breadth as a function of the length of the vessel.


Waterplane Area (2.9.1)

• Writing out the Simpson’s equation:

where:

Awp is the waterplane area in ft2

s is the Simpson’s spacingy(x) is the “y” offset or half-breadth at each value of “x” in ft

Example for a ship!


0 2.5 5 7.5 102 ( 4 2 4 )3 stn stn stn stn stn

sAwp y y y y y

Section Area (2.9.2)

• Numerical integration of the half- breadth as a function of the draft.

To Port

Deck

WL@10'

WL@5'

Baseline

Up

Ex. View atSection 5,looking aft.This is fromthe Body Plan.

Stn 5


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:


sec 2.5 12.5 2.5 5 7.5 10 12.5

sec 0 2.5 2.5

2 ( 4 2 4 )3

2*0.5* *

_ _ _ _

twl wl wl wl wl wl

twl wl

sA y y y y y

A s y

then add the two areas


• Archimedes Principle - The Equation

gFB

Recall that the goal of us using the Lines PlanAnd the Table of Offsets was to find the Volume…, and hence the buoyant force!

And, if in static equilibrium, then FB=Weight!But so far, we can only calculate the section and waterplane areas…

Submerged Volume: Longitudinal Integration (2.9.3)

• Integration of the section areas over the length of the ship. “Curve of Areas”


dx

ASect(x)

z

y

x

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


What is a barge’s section area, volume, curve of areas and displacement?

Section Area = Beam x DraftVolume = Section Area x Length100 ft long, 25 feet beam and 10 feet draft

FP AP

Curve of Areas

SectionArea

gFB

Submerged Volume: Longitudinal Integration (2.9.3)


So, the volume if using Simpson’s is:

)AA2A4A(3

sVol 10210s

Ques: where is the “2”?

Longitudinal Center of Flotation (LCF) (2.9.4)(Centroid of Waterplane Area)


• 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.



• Drawing of the LCF:

y

FP AP xdx

y(x)

XLCF

Recall: For most “normal” vessels LCF is between Stn 5 and 6.7



• Writing the general calculus equation and the general Simpson’s form (for 4 Simpson’s spaces in a 10 station ship):

0

0 0 2.5 2.5 5 5 7.5 7.5 10 10

2( )

2( 4 2 4 )

3

LWL

stn stn stn stn stn stn stn stn stn stn

LCF x y x dxAwp

sLCF x y x y x y x y x y

Awp

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

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.


• Curves of Form Include:

– Displacement– LCB– VCB– Immersion (TPI)– LCF– MT1– And some others...

Curves of Form (2.10.1.2)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!


• 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.


• 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.


• An approximate formula for TPI based on the area of the waterplane can be derived as follows:

23

2

( )

1

( )

1

1 6412 2240

1

( )420

weight to increasethedraft oneinch LTTPI

in

g for oneinch

in

lb ft LTAwp ft in

ft in lbin

Awp LTft

in


• 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.


• 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?


• 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.


• 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.


• KML : (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.


• KMT: (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.

The End of Chapter 2

Did you meet all the chapter’s objectives?!

In one word… buoyancy!

chapter 2

Documents

geometry basic

geometry categorizing

centerline

barges section

geometry waterplane

geometry hull

hydrostatic

simpsons rule