HULL FORM AND GEOMETRY Chapter 2 Intro to Ships and Naval Engineering (2.1) Factors which influence design: Size Speed Payload Range

Seakeeping Maneuverability Stability Special Capabilities (Amphib, Aviation, ...) Compromise is required! Classification of Ship by Usage Merchant Ship Naval & Coast Guard Vessel

Recreational Vessel Utility Tugs Research & Environmental Ship Ferries Categorizing Ships (2.2) Methods of Classification: Physical Support: Hydrostatic

Hydrodynamic Aerostatic (Aerodynamic) Categorizing Ships Classification of Ship by Support Type Aerostatic Support - ACV - SES (Captured Air Bubble) Hydrodynamic Support (Bernoulli) - Hydrofoil - Planning Hull Hydrostatic Support (Archimedes) - Conventional Ship - Catamaran - SWATH - Deep Displacement Submarine - Submarine - ROV

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 Aerostatic Support Supported by cushion of air ACV hull material : rubber propeller : placed on the deck amphibious operation SES side hull : rigid wall(steel or FRP) bow : skirt propulsion system : placed under the water

water jet propulsion supercavitating propeller (not amphibious operation) Aerostatic Support Aerostatic Support English Channel Ferry - Hovercraft Aerostatic Support SES Ferry NYC SES Fireboat E 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. Hydrodynamic Support Planing Hull - supported by the hydrodynamic pressure developed under a hull at high speed - V or flat type shape - Commonly used in pleasure boat, patrol boat, missile boat, racing boat Destriero Hydrodynamic Support Hydrofoil Ship

- supported by a hydrofoil, like wing on an aircraft - fully submerged hydrofoil ship - surface piercing hydrofoil ship Hydrofoil Ferry Hydrodynamic Support Hydrodynamic Support Hydrostatic Support 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 (when surfaced)

Hydrostatic Support The Ship is supported by its buoyancy. (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. The buoyant force of a ship is calculated from the displaced volume by the ship. Hydrostatic Support Mathematical Form of Archimedes Principle Resultant Weight S FB g FB : Magnitude of the resultant buouant force(lb)

: Density of fluid (lb s2 /ft 4 ) g : Gravitational accelerati on(32.17ft/s) : Displaced volume by the object(ft 3 ) Resultant Buoyancy FB F B S Hydrostatic Support Displacement ship - conventional type of ship - carries high payload - low speed SWATH - small waterplane area twin hull (SWATH) - low wave-making resistance - excellent roll stability

- large open deck - disadvantage : deep draft and cost Catamaran/Trimaran - twin hull - other characteristics are similar to the SWATH Submarine Hydrostatic Support Hydrostatic Support Hydrostatic Support Hydrostatic Support Hydrostatic Support Hydrostatic Support Hydrostatic Support

Hydrostatic Support 2.3 Ship Hull Form and Geometry The ship is a 3-dimensional shape: Data in x, y, and z directions is necessary to represent the ship hull. Table of Offsets Lines Drawings: - body plan (front View) - shear plan (side view) - half breadth plan (top view) Hull Form Representation Lines Drawings: Traditional graphical representation of the ships hull form Lines Half-Breadth Sheer Plan

Body Plan Hull Form Representation Body Plan (Front / End) Half-Breadth Plan (Top) Lines Plan Sheer Plan (Side) Half-Breadth Plan - Intersection of planes (waterlines) parallel to the baseline (keel). F ig u r e

2 .3 - T h e H a lf - B r e a d th P la n Sheer Plan -Intersection of planes (buttock lines) parallel to the centerline plane F i g u r e 2 . 4 - T h e S h e e r P la n Body Plan - Intersection of planes to define section line - Sectional lines show the true shape of the hull form - Forward sections from amidships : R.H.S. - Aft sections from amid ship : L.H.S.

F i g u r e 2 . 6 - T h e B o d y P la n 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.

Table of Offsets The distances from the centerplane are called the offsets or half-breadth distances. 2.5 Basic Dimensions and Hull Form Characteristics AP FP Shear DWL Lpp LOA LOA(length over all ) : Overall length of the vessel DWL(design waterline) : Water line 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. FP(forward perpendicular) : imaginary vertical line where the bow intersects the DWL AP(aft perpendicular) : imaginary vertical line located at either the rudder

stock or intersection of the stern with DWL Basic Dimensions and Hull Form Characteristics AP FP Shear DWL Lpp LOA Lpp (length between perpendicular) : horizontal distance from FP and AP Amidships : the point midway between FP and AP ( Station ) Midships Basic Dimensions and Hull Form Characteristics Beam: B

View of midship section WL Camber Freeboard Depth: D Draft: T K CL Depth(D): vertical distance measured from keel to deck taken at amidships and deck edge in case the ship is cambered on the deck. Draft(T) : vertical distance from keel to the water surface Beam(B) : transverse distance across the each section Breadth(B) : transverse distance measured amidships Basic Dimensions and Hull Form Characteristics

View of midship section Beam: B Camber Freeboard WL Depth: D Draft: T K C L Freeboard : distance from depth to draft (reserve buoyancy) Keel (K) : locate the bottom of the ship Camber : transverse curvature given to deck

Basic Dimensions and Hull Form Characteristics Flare Tumblehome Flare : outward curvature of ships hull surface above the waterline Tumble Home : opposite of flare Example Problem Label the following: R. Distance between N. & O. ___=______ _______ ______________ I. Viewed from P. Middle ref plane for longitudinal measurements this direction ____-_______ Plan _________

S. Width of the ship ____ E. (rotation) _____/____ Q. Longitudinal ref plane for transverse measurements __________ J. _______ Line O. Aft ref plane for longitudinal measurements ___ _____________ D. (rotation) ____/____/____ z C. (translation) _____

y L. _____line K. _______ Line G. Viewed from this direction ____ Plan F. (rotation) ___ x A.(translation) _____ N. Forward ref plane for longitudinal measurements _______ _____________

M. Horizontal ref plane for vertical measurements ________ H. Viewed from this direction _____ Plan B. (translation) ____ Example Answer Label the following: R. Distance between N. & O. LBP=Length between Perpendiculars I. Viewed from G. Viewed from P. Middle ref plane for

this direction longitudinal measurements this direction Half-Breadth Plan Body Plan Amidships z S. Width of the ship A.(translation) Beam Surge E. (rotation) C. (translation) x Pitch/Trim Heave Q. Longitudinal ref plane for N. Forward ref plane for transverse measurements longitudinal measurements Centerline Forward Perpendicular

J. Section Line O. Aft ref plane for longitudinal measurements Aft Perpendicular D. (rotation) Roll/List/Heel y L. Waterline K. Buttock Line F. (rotation) Yaw M. Horizontal ref plane for vertical measurements Baseline H. Viewed from this direction

Sheer Plan B. (translation) Sway 2.6 Centroids Centroid - Area - Mass - Volume - Force - Buoyancy(LCB or TCB) - Floatation(LCF or TCF) Apply the Weighed Average Scheme or Moment =0 Centroids 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! Centroids Centroids and Center of Mass can be found by using a weighted average. Y a1 a2 a3 an y ave y a

a i 1 i 1 y1 y2 y3 y ave i i i

yn X y 1a y 2a 2 y 3a 3 a1 a 2 a 3 1 Centroid of Area y a1 x

a3 a2 x2 y x1 y1 y2 y3 x3 x n xa

i i x i 1 AT n n ai xi i 1 AT ya

i i y i 1 xi : distance from y - axis to differential area center y i : distance from x - axis to differential area center ai : differential area A T a1 a 2 a n AT n ai yi i 1 AT

Centroid of Area Example y 5ft 3ft x 4 y 2 2 8ft 2

3 7 x n xa 2 2 2 3 a 3 ft

2 ft 5 ft 4 ft 8 ft 7 ft 82 ft x i 1 xi i 2

2 2 2 AT A 3 ft 5 ft 8 ft 16 ft i 1 T 5.125 ft from y - axis i i

n 3 ya i i y i 1 AT 3 ai yi ..... i 1 AT

Centroid of Area y Proof b xdA x x1 AT x x1 b

AT h x x1 dx x 1 2 hb hbx1 x hdx 2 AT hb 1 x1 b

2 Since the moment created by differential area dA is dM xdA , total moment which is called 1st Moment of Area is calculated by integrating the whole area as, M xdA Also moment created by total area AT will produce a moment w.r.t y axis and can be written below. (recall Moment=forcemoment arm) M AT x The two moments are identical so that centroid of the geometry is xdA x AT This eqn. will be used to determine LCF in this Chapter.

2.7 Center of Floatation & Center of Buoyancy Center of Floatation - Centroid of water plane (LCF varies depending on draft.) - Pivot point for list and trim of floating ship LCF: centroid of water plane from the amidships TCF : centroid of water plane from the centerline The Center of Flotation changes as the ship lists, trims, or changes draft because as the shape of the waterplane changes so does the location of the centroid. LCF centerline TCF Amidships In this case of ship, - LCF is at aft of amidship. - TCF is on the centerline.

Center of Buoyancy - Centroid of displaced water volume - Buoyant force act through this centroid. LCB: Longitudinal center of buoyancy from amidships KB : Vertical center of buoyancy from the Keel TCB : Transverse center of buoyancy from the centerline Center of Buoyancy moves when the ship lists, trims or changes draft because the shape of the submerged body has changed thus causing the centroid to move. LCB TCB KB Center line Base line

Center of Buoyancy : B B centerline 2 1 WL 1 2 - 2 Buoyancy force (Weight of Barge) LCB : at midship TCB : on centerline KB : T/2

Reserve Buoyancy Force 1 1 1 1 WL B CL T/2 2.8 Fundamental Geometric Calculation Why numerical integration? - Ship is complex and its shape cannot usually be represented by a mathematical equation. - A numerical scheme, therefore, should be used to calculate the ships

geometrical properties. - Uses the coordinates of a curve (e.g. Table of Offsets) to integrate Which numerical method ? - Rectangle rule - Trapezoidal rule - Simpsons 1st rule (Used in this course) - Simpsons 2nd rule Rectangle rule Trapezoidal rule Simpsons rule Trapezoidal Rule - Uses 2 data points - Assumes linear curve A1

x1 y4 y2 y1 s y3 A2 x2 : y=mx+b A3 A1=s/2*(y1+y2)

A2=s/2*(y2+y3) A3=s/2*(y3+y4) s x3 s x4 s = x = x2-x1 = x3-x2 = x4-x3 Total Area = A1+A2+A3 = s/2 (y1+2y2+2y3+y4) Simpsons 1st Rule - Uses 3 data points - Assume 2nd order polynomial curve Mathematical Integration y Numerical Integration y(x)=ax+bx+c

dx dA x2 x3 x y2 y3 x x1 s x2 s x3 (S=x)

x3 Area : y1 A A x1 y A dA x1 x y dx ( y1 4 y2 y3 ) 3

Simpsons 1st Rule y y2 y1 y3 s x1 x2 x3 y4 y6 y7 y8 y9

y5 x x4 x5 x6 x7 x8 x9 s s A ( y1 4 y 2 y 3 ) ( y 3 4 y 4 y 5 ) Odd number 3 3 Evenly spaced s s ( y5 4 y6 y7 ) ( y7 4 y8 y9 ) 3 3 s ( y1 4 y 2 2 y 3 4 y 4 2 y 5 4 y 6 2 y 7 4 y 8 y 9 ) 3

x A ( y 1 4 y 2 2 y 3 ... 2 y n 2 4 y n 1 y n ) Gen. 3 Application of Numerical Integration Application - Waterplane Area - Sectional Area - Submerged Volume - LCF - VCB - LCB Scheme - Simpsons 1st Rule 2.9 Numerical Calculation Calculation Steps

1. Start with a sketch of what you are about to integrate. 2. Show the differential element you are using. 3. Properly label your axis and drawing. 4. Write out the generalized calculus equation written in the same symbols you used to label your picture. 5. Convert integral in Simpsons equation. 6. Solve by substituting each number into the equation. Section 2.9 See your Equations and Conversions Sheet Y (Half-Breadth Plan) y(x) HalfBreadths (feet) Waterplane Area

dx=Station Spacing 0 X Stations AWP=2y(x)dx; where integral is half breadths by station Sectional Area Z Asect=2y(z)dz; where integral is half breadths by waterline 0

Water lines (Body Plan) dz=Waterline Spacing y(z) 0 Half-Breadths (feet) Y Section 2.9 See your Equations and Conversions Sheet Submerged Volume VS=Asectdx; where integral is sectional areas by station Asect A(x)

Sectional Areas (feet) dx=Station Spacing 0 (Half-Breadth Plan) y(x) Y Longitudinal Center of Floatation LCF=(2/AWP)*xydx; where integral is product of 0distance from FP & half breadths by station

HalfBreadths (feet) X Stations dx=Station Spacing x Stations X Waterplane Area y y(x) x FP

AWP 2 dx Lpp dA 2 0 AP y( x ) dx area AWP waterplane area ( ft 2 ) Factor for symmetric waterplane area dA differential area ( ft 2 ) y( x) y offset (half - breadth) at x (ft)

dx differential width (ft) Waterplane Area Generalized Simpsons Equation y x x FP 0 AWP 1 2 3 4

5 6 AP 1 2 x y 0 4 y1 2 y 2 .. 2 y n 2 4 y n 1 y n 3 x distance between stations Sectional Area Sectional Area : Numerical integration of half-breadth as a function of draft z WL T y(z)

Asect 2 T dA 2 y( z) dz area 0 Asec t sectionalarea up to z ( ft 2 ) dz y dA differential area( ft 2 ) y( z ) y offset(half - breadth)at z( ft ) dz differential width( ft )

Sectional Area Generalized Simpsons equation z T WL 8 6 4 2 0 Asect 2 z z distance btwn waterlines

y T dA 2 y( z) dz area 0 1 2 z y0 4 y1 2 y2 .. 2 yn 2 4 yn 1 yn 3 Submerged Volume : Longitudinal Integration Submerged Volume : Integration of sectional area over the length of ship z Scheme:

x As (x ) y Submerged Volume Sectional Area Curve As Asec t ( x ) dx FP Calculus equation x AP L pp

Vsubmerged s dV A volume sect ( x)dx 0 Generalized equation 1 s x y0 4 y1 2 y2 .. 4 yn 1 yn 3 x distance between stations

Asection, Awp , and submerged volume are examples of how Simpsons rule is used to find area and volume The next slides show how it can be used to find the centroid of a given area. The only difference in the procedure is the addition of another term, the distance of the individual area segments from the y-axisthe value of x. The values of x will be the progressive sum of the x if x is the width of the sections, say 10, then x0=0, x1=10, x2=20,x3=30 and so on. Longitudinal Center of Floatation(LCF) LCF - Centroid of waterplane area - Distance from reference point to center of floatation - Referenced to amidships or FP - Sign convention of LCF + -

WL + FP Longitudinal Center of Floatation (LCF) y dA y(x) x FP dx AP

Weighted Average of Variable X (i.e. distance from FP) Average of variable X all X Moment Relation x 2 xdA AWA small piece X value total First moment of area : M y xdA

2 xy ( x )dx AWA Recall xdA xy ( x )dx x AT AT Longitudinal Center of Floatation(LCF) y y(x) LCF

FP dx x AP LCF by weighted averaged scheme or Moment relation xdA Lpp 2 xy( x) LCF dx 0 A AWP WP area 2

AWP Lpp 0 x y( x) dx Longitudinal Center of Floatation(LCF) Generalized Simpsons Equation x6 x5 x4 y x3 x x1 2 x FP

0 1 2 LCF AWP 2 3 x 4 5 6

AP L pp x y ( x ) dx x0 0, x1 x, x2 2x, x3 .... 0 2 1 x x0 y0 4 x1 y1 2 x2 y2 .. 4 xn 1 yn 1 xn yn A WP 3 x distance between stations Its often easier to put all the information in tabular form on an Excel spreadsheet: Station

Dist from FP (x value) 0 1 2 3 4 0.0 81.6 163.2 244.8 326.4 HalfBreadth (y value) 0.39 12.92

20.97 21.71 12.58 Moment x y 0.0 1054.3 3422.3 5314.6 4106.1 Simpson Multiplier Product of Moment x Multiplier 1

4 2 4 1 Remember, this gives only part of the equation! .You still need the 2/Awp x 1/3 Dx part! Dx here is 81.6 ft Awp would be given 2 because youre dealing with a half-breadth section 0.0 4217.1 6844.6 21258.4 4106.1 36426.2 Vertical Center of Buoyancy, KB This is similar to the LCF in that it is a CENTROID, but where LCF is the centroid

of the Awp, KB is the centroid of the submerged volume of the ship measured from the keel z y Awp KB x zA KB WP ( z )dz

where: - Awp is the area of the waterplane at the distance z from the keel - z is the distance of the Awp section from the x-axis - dz is the spacing between the Awp sections, or Dz in Simpsons Eq. You can now put this into Simpsons Equation: zA KB WP ( z )dz KB =1/3 dz [(1) (zo) (Awpo) + 4 (z1) (Awp1) + 2 (z2) (Awp2) + + (zn) (Awpn) ]/

underwater hull volume Remember that the blue terms are what we have to add to make Simpson work for KB. Dont forget to include them in your calculations! And FINALLY, Longitudinal Center of Buoyancy, LCB This is EXACTLY the same as KB, only this time: - Instead of taking measurements along the z-axis, youre taking them from the x-axis - Instead of using waterplane areas, youre using section areas - Itll tell you how far back from the FP the center of buoyancy is. z y Asection x LCB

xA LCB Sect ( x)dx where: - Asect is the area of the section at the distance z from the forward perpendicular (FP) - x is the distance of the Asect section from the y-axis - dx is the spacing between the Asect sections, or Dx in Simpsons Eq. You can now put this into Simpsons Equation: xA

LCB Sect ( x)dx LCB = 1/3 dx [(1) (xo) (Asect) + 4 (x1) (Asect 1) + 2 (x2) (Asect 2) + + (xn) (Asect n) ] underwater hull volume Remember that the blue terms are what we have to add to make Simpson work for LCB. Dont forget to include them in your calculations! / And that is Simpsons Equations as they apply to this course! Remember:

The concept of finding the center of an area, LCF, or the center of a volume, LCB or KB, are just centroid equations. Understand THAT concept, and you can find the center of any shape or object! And: Dont waste your time memorizing all the formulas! Understand the basic Simpsons 1st, understand the concept behind the different uses, and youll never be lost! 2.10 Curves of Forms Curves of Forms A graph which shows all the geometric properties of the ship as a function of ships mean draft Displacement, LCB, KB, TPI, WPA, LCF, MTI, KML and KMT are usually included. Assumptions Ship has zero list and zero trim (upright, even keel) Ship is floating in 59F salt water Curves of Forms

Displacement ( ) - assume ship is in the salt water with - unit of displacement : long ton 1 long ton (LT) =2240 lb 1.99 (lb s2 /ft 4 ) LCB - Longitudinal center of buoyancy - Distance in feet from reference point (FP or Amidships) VCB or KB - Vertical center of buoyancy - Distance in feet from the Keel Curves of Forms TPI (Tons per Inch Immersion) - TPI : tons required to obtain one inch of parallel sinkage in salt water - Parallel sinkage: the ship changes its forward and aft draft by the same amount so that no change in trim occurs

- Trim : difference between forward and aft draft of ship Trim Taft Tfwd - Unit of TPI : LT/inch Note: for parallel sinkage to occur, weight must be added at center of flotation (F). TPI 1 inch Awp (sq. ft) 1 inch - Assume side wall is vertical in one inch. - TPI varies at the ships draft because waterplane area changes at the draft Curves of Forms

weight required for one inch 1 inch Volume required for one inch salt g 1 inch Awp ( ft 2 )(1 inch) 1.99lb * s 2 / ft 4 32.17 ft / s 2 1 ft 1 LT 1 inch 12 inches 2240 lb TPI Awp ( ft 2 ) LT 420 inch

1 inch Awp Curves of Forms Change in draft due to parallel sinkage w Tps TPI Tps change in draft (inches) w amount of weight added or removed (LT) Curves of Forms Moment/Trim 1 inch (MT1) - MT1 : moment to change trim one inch - The ship will rotate about the center of flotation when a moment is applied to it. - The moment can be produced by adding, removing or shifting a weight some distance from F. - Unit : LT-ft/inch

FP AP wl Trim MT 1" l W F Change in Trim due to a Weight Addition/Removal 1 inch Curves of Forms - When MT1 is due to a weight shift, l is the distance the weight was moved

- When MT1 is due to a weight removal or addition, l is the distance from the weight to F LCF l W1 W0 New waterline Curves of Forms KM L - Distance in feet from the keel to the longitudinal metacenter KM T

- Distance in feet from the keel to the transverse metacenter M M B K KM T B AP K KM L FP

Example Problem A YP has a forward draft of 9.5 ft and an aft draft of 10.5ft. Using the YP Curves of Form, provide the following information: = _____ KMT=____ WPA= _____ LCF= _____ TPI=____ LCB=____ VCB=____ KML=____ MT1=_________ Example Answer

A YP has a forward draft of 9.5 ft and an aft draft of 10.5ft. Using the YP Curves of Form, provide the following information: = 192.52 LT = 385 LT KMT = 192.5.06 ft = 11.55 ft WPA = 2358.4 ft = 1974 ft LCF = 56 ft fm FP LCB = 56 ft fm FP VCB = 125.05 ft = 6.25 ft TPI = 235.02 LT/in = 4.7 LT/in KML = 1121 ft = 112 ft MT1 = 250.141 ft-LT/in = 35.25 ft-LT/in Backup Slides Example Problem A 40 foot boat has the following Table of Offsets (Half Breadths in Feet): Half-Breadths from Centerline in Feet

Station Numbers WATERLINE FP (ft) 0 1 4 1.1 5.2 2 8.6 3 10.1 AP 4 10.8

What is the area of the waterplane at a draft of 4 feet? Example Answer Y Half-Breadths at 4 Foot Waterlines y(x) HalfBreadths (Feet) 0 Station Spacing=dx =40ft/4=10ft Station 4

AWP=2y(x)dx ydx=s/3*[1y0+4y1++2yn-2+4yn-1+1yn] AWP=2*10ft/3*[1(1.1ft)+4(5.2ft)+2(8.6ft)+4(10.1ft) +1(10.8ft)] AWP=602ft X Simpsons Rule Simpsons Rule is used when a standard integration technique is too involved or not easily performed. A curve that is not defined mathematically A curve that is irregular and not easily defined mathematically It is an APPROXIMATION of the true integration Given an integral in the following form: y

y ( x ) dx y = f(x) x Where y is a function of x, that is, y is the dependent variable defined by x, the integral can be approximated by dividing the area under the curve into equally spaced sections, Dx, y y = f(x) Dx and summing the individual areas.

y y = f(x) x Notice that: Dx Spacing is constant along x (the dx in the integral is the Dx here) The value of y (the height) depends on the location on x (y is a function of x, aka y= f(x) The area of the series of rectangles can be summed up Simpsons Rule breaks the curve into these sections and then sums them up for total area under the curve Simpsons 1st Rule Area = 1/3 Dx [yo + 4y1 + 2y2+2y n-2 + 4y n-1 + yn] where:

- n is an ODD number of stations - Dx is the distance between stations - yn is the value of y at the given station along x - Repeats in a pattern of 1,4,2,4,2,4,22,4,1 Simpsons 2nd Rule Area = 3/8 Dx [yo + 3y1 + 3y2 + 2y3 + 3y4 +3y5 + 2y6 + + 3y n-1 + yn] where: - n is an EVEN number of stations - Repeats in a pattern 1,3,3,2,3,3,2,3,3,2,2,3,3,1 Simpsons 1st Rule is the one we use here since it gives an EVEN number of divisions Heres how its put to use in this course: Waterplane Area, Awp AWP 2y ( x)dx Awp = 2 x 1/3 Dx [yo + 4y1 + 2y2+2y n-2 + 4y n-1 + yn]

The 2 is needed because the data youll have is for a half-section Section Area, Asect Asect 2 y ( z )dz Asect = 2 x 1/3 Dx [yo + 4y1 + 2y2+2y n-2 + 4y n-1 + yn] Note: You will always know the value of y for the stations (x or z)! It will be presented in the Table of Offsets or readily measured Simpsons 1st Rule - Uses 3 data points - Assume 2nd order polynomial curve Mathematical Integration y Numerical Integration y(x)=ax+bx+c

dx dA x2 y2 y3 x3 x x1 s x2 s x3 x x3

Area : y1 A A x1 y A dA x1 s y dx ( y1 4 y2 y3 ) 3 Simpsons 1st Rule

y y2 y1 s x1 y3 y4 y6 y7 y8 y9 y5 s x2 x3

x4 x5 x6 x7 x8 x9 x s s A ( y1 4 y 2 y 3 ) ( y 3 4 y 4 y 5 ) Odd number 3 3 s s ( y5 4 y6 y7 ) ( y7 4 y8 y9 ) 3 3 s ( y1 4 y 2 2 y 3 4 y 4 2 y 5 4 y 6 2 y 7 4 y 8 y 9 )

3 s A ( y 1 4 y 2 2 y 3 ... 2 y n 2 4 y n 1 y n ) Gen. 3 We can now move onto the next dimension, VOLUMES! Volume, Submerged, Vsubmerged - It gets a little trickier here remember, since you are now dealing with a VOLUME, the y term previous now becomes an AREA term for that station section because you are summing the areas: Vsubmerged Asect ( x) dx Vsub = 1/3 Dx [Ao + 4A1 + 2A2+2A n-2 + 4A n-1 + An] Simpsons 2nd Rule - uses 4 data points - assumes 3rd order polynomial curve

y y2 y1 y4 y3 y(x)=ax+bx+cx+d A x1 s x2 s x3

x4 x 3 s ( y 3 y 3 y y ) Area : A 1

2 3 4 8 Longitudinal Center of Flotation, LCF - This is the CENTROID of the Awp of the ship. - For this reason you now need to introduce the distance, x, of the section Dx from the y-axis y y(x) dA x FP Dx

AP LCF 2 / AWP xdA That is, LCF is the sum of all the areas, dA, and their distances from the y-axis, divided by the total area of the water plane Longitudinal Center of Flotation, LCF, (contd) - Since our sectional areas are done in half-sections this needs to be multiplied by 2 - Remember, dA = y(x)dx, so we can substitute for dA - Awp is constant, so it moves left dA LCF =2/Awp x dA 2/Awp x y(x)dx Substituting into Simpson's Eq., youll get the following:

LCF = 2/Awp x 1/3 Dx [(1) (xo) (yo) + 4 (x1) (y1) + 2 (x2) (y2) + + (xn) (yn) ] Note that the blue terms are what we have to add to make Simpson work for LCF. Remember to include them in your calculations!