HYDROSTATICS: Hull Geometric Calculations

Fundamental Hull Geometric C l l tiCalculations

Numerical methods are used in order to calculate the Numerical methods are used in order to calculate the fundamental geometric properties of the hull

The trapezoidal rule and Simpson's Rule are two methods of numerical calculation frequently used.

Numerical Calculations involved such as Waterplane Area Sectional Area Submerged Volume LCF LCBArea, Sectional Area, Submerged Volume, LCF, LCB and VCB

Moreover, all hydrostatic particulars will be calculated y pusing this approach.

Trapezoidal MethodTh i d t The curve is assumed to be represented by a set of trapezoidsof trapezoids.

The area under the curve is the area of total trapezoid ABCDEFis the area of total trapezoid ABCDEF

Area=

Simpson Rule the most popular and common method being used in

naval architecture calculations It is flexible, easy to use, its mathematical basis is easily

understood, greater accuracy, and the result reliable., g y, Its rule states that ship waterlines or sectional area

curves can be represented by polynomials Using calculus the areas volumes centroids and Using calculus, the areas, volumes, centroids and

moments can be calculated from these polynomials With Simpson rules, the calculus has been simplified by

i lti l i f t lti liusing multiplying factors or multipliers. There are 3 Simpson rules, depending on the number

and location of the offsets.

1. Simpson 1st RuleU d h th i dd Used when there is an odd number of offset

The basic multiplier for set of The basic multiplier for set of three offsets are 1, 4, 1

The multiplier must begin p gand end with 1

For more stations (odd b ) th lti linumbers), the multipliers

become 1,4,2,4,2……4,1 This can be proved asThis can be proved as

follows:

where y is a offset distanceh is a common interval

Area= )(31 offsetmultiplierh

2. Simpson 2nd Rule Only can be used when number of offsets = 3N+1

(N i b f ff t)(N is number of offset) The basic multiplier for set of four offsets are 1, 3, 3, 1 The multiplier also must begin and end with 1 The multiplier also must begin and end with 1 For more stations , the multipliers become

1,3,3,2,3,3,2……,3,3,1

Also the area is preferable to be written as:Area = )(3 ffl i lih Area = )(8

3 offsetmultiplierh

3. Simpson 3rd Rule Commonly known as the 5,8-1 rule. This is to be used when the area between any two

adjacent ordinates is required, three consecutive ordinates being givenordinates being given.

The multipliers are 5,8,-1.

Obtaining AreaObtaining Area Area is the first important geometry that need to be

calculated.2 t f W t l WPA d 2 common types of area, Waterplane area, WPA and Sectional Area, AS (or sometimes known as Station Area).

Waterplane area, WPA has its centroid called longitudinal centre floatation (LCF)

LCF need to be determined for various waterplane areas, WPA (at various waterlines)WPA (at various waterlines)

In overall for applying the Simpson method it is more

Waterplane area, WPA Sectional Area, AS

In overall, for applying the Simpson method, it is more comfortable making a tables in solving the calculation

A B C D E F G H

S i ½ di SM P d L P d L P dStation ½ ordinate SM Product Area

Lever Product 1st mmt

Lever Product 2nd mmt

ΣProduct Area

Σ Product 1st mmt

Σ Product 2nd mmt

Waterplane area, WPA= 1/3 x Σproduct area x h

1st moment = 1/3 x Σproduct 1stmmt x h x h

hhd3/1LCF = hproduct

hhproduct

area

mmtst

3/13/1 1

e.g. for 1st

Simpson Rulehproduct st

=

2nd moment, IL = (1/3 x Σproduct 2ndmmt x h x h2) x 2

area

mmt

producthproduct st

1

, L ( p 2 mmt )

*h = common interval (in this case, station spacing)

Lever is set accordingly to the desired reference point Lever is set accordingly to the desired reference point (datum point). It can be set either zero at aft, amidship or forward of the ship.

If reference point is set atAftp p

For example;Station ½ ordinate SM Product

AreaOption1 Option 2

LeverOption 3

LeverProduct 2nd mmt

( )

AmidshipForward

Area Lever Lever Lever (Product Area x Lever)

AP 1.1 1 1.1 0 -3 6

1 2.7 4 10.8 1 -2 5

2 4 0 2 8 0 2 1 42 4.0 2 8.0 2 -1 4

3 5.1 4 20.4 3 0 3

4 6.1 2 12.2 4 1 2

5 6.9 4 27.6 5 2 15 6.9 4 27.6 5 2 1

FP 7.7 1 7.7 6 3 0

ΣProduct Area

Σ Product 2nd mmt

Exercise 1For a supertanker, her fully loaded waterplane o a supe a e , e u y oaded a e p a ehas the following ½ ordinates spaced 45m apart:p0, 9.0, 18.1, 23.6, 25.9, 26.2, 22.5, 15.7 and 7.2 metres respectively.p yCalculate the waterplane area, WPA and waterplane area coefficient, Cwp.p , p

Exercise 2A water plane of length 270m and breadth 35.5m p ghas the following equally spaced breadth 0.3, 13.5, 27.0, 34.2, 35.5, 35.5, 32.0, 23.1 and 7.4 m

ti lrespectively.Calculate;1.Waterplane area, WPA, and its coefficient, Cwp2.Longitudinal Centre of Floatation, LCF about the

id hiamidships.3.Second moment of area about the amidships

Obtaining VolumeVolumes, hence displacement of the ship at any draught can beany draught can be calculated if we know either;i) Waterplane areas at

i t li tWL 2

WL 3

various waterlines up to required draught, ORii) Sectional areas up to the

Waterplane areas at various waterlinesWL 1

ii) Sectional areas up to the required draught at various stationsVolume has its centroid, called longitudinal centre of buoyancy (LCB) and vertical y y ( )centre of buoyancy (VCB)

Sectional areas at various stations

A B C D E F

Station Station Area

SM Product Volume

Lever Product 1st mmt

ΣProduct Volume

Σ Product 1st mmt

3Volume Displacement, (m3)= 1/3 x Σproduct volume x h

Displacement, ∆ (tonne)= Volume Displacement x ρ

1st moment = 1/3 x Σproduct 1stmmt x h x h

LCB = hproduct st 1

e.g. for 1st Simpson Rule

LCB =

volume

mmt

producthp oduct st

1

*h = common interval(in this case, waterline spacing)

ExampleExample

S ti l f 180 LBP hi t 5 Sectional areas of a 180m LBP ship up to 5m draught at constant interval along the length are as follows Find its volume displacement and its LCBfollows. Find its volume displacement and its LCB from amidships.

Station 0 1 2 3 4 5 6 7 8 9 10Station 0 1 2 3 4 5 6 7 8 9 10

Area ( 2)

5 118 233 291 303 304 304 302 283 171 0(m2)

ExampleA ship length of 150m, breadth 22m has the s p e g o 50 , b ead as efollowing waterplane areas at various draught. Find the volume, displacement volume and pvertical centre of buoyancy, VCB at draught 10m

Draught (m) 2 4 6 8 10

Waterplane area, WPA (m2)

1800 2000 2130 2250 2370

HYDROSTATICS (part II): Hydrostatics Particulars and y

Curves

Displacement (Δ) This is the weight of the water displaced by the ship for a given draft assuming the ship is in salt water with a density of 1025kg/m3.

LCB This is the longitudinal center of buoyancy. It is the distance g y yin feet from the longitudinal reference position to the center of buoyancy. The reference position could be the AP, FP or midships If it is midships remember that distances aft ofmidships. If it is midships remember that distances aft of midships are negative.

VCB This is the vertical center of buoyancy It is the distance inThis is the vertical center of buoyancy. It is the distance in meter from the baseline to the center of buoyancy. Sometimes this distance is labeled KB.

WPA or Aw WPA or Aw stands for the waterplane area. The units of WPA are m2 It can be calculated using Simpson RuleWPA are m2. It can be calculated using Simpson Rule

LCF LCF is the longitudinal center of flotation. It is the distance in from the longitudinal reference to the center of flotation. The reference position could be the AP, FP or amidships. If it is midships remember that distances aft of amidships areit is midships remember that distances aft of amidships are negative.

Immersion or TPCImmersion or TPC TPC stands for tonnes per centre meter or sometimes just called immersion. TPC is defined as the tonnes required to obtain one centre meter of parallel sinkage in salt water. P ll l i k i h th hi h it’ f d dParallel sinkage is when the ship changes it’s forward and after drafts by the same amount so that no change in trim occurs.

SWWATPC

100

MCTCTo show how easy a ship is to trim The value in SI unitsTo show how easy a ship is to trim. The value in SI units would be moment to change trim one centre meter.Trim is the difference between draught forward and aft. The excess draught aft is called trim by the stern, while at forward is called trim by the bow

GML

GMMCTC L

100

KML KML This stands for the distance from the keel to the longitudinal metacenter. For now just assume the metacenter is a convenient reference point vertically abovemetacenter is a convenient reference point vertically above the keel.

KML= KB + BML

LCF

LIBM

22)( midshipmidshipLCF LCFWPAII

KMT

This stands for the distance from the keel to the transverseThis stands for the distance from the keel to the transverse metacenter. Typically, Naval Architects do not bother putting the subscript “T” for any property in the transverse directiondirection.

KMT = KB + BMT

A B C D E

Station ½ ordinate (½ ordinate)3

SM Product 2nd mmtKMT = KB + BMT

TIBM

ordinate) 2 mmt

ΣProduct

TBM ΣProduct 2nd mmt

2231

31 mmtproducthI nd

T 33 pT

e.g. is applicable for 1st Simpson Rule

H d t ti CHydrostatic Curves All the geometric properties of a ship as a function of

mean draft have been computed and put into a singlemean draft have been computed and put into a single graph for convenience.

This graph is called the “curves of form” or Hydrostatic CurvesCurves.

Each ship has unique curves of form. There are also tables with the same information which are called the tabular curves of form or Hydrostatic Tabletabular curves of form, or Hydrostatic Table.

It is difficult to fit all the different properties on a single sheet because they vary so greatly in magnitude.

The curves of form assume that the ship is floating on an The curves of form assume that the ship is floating on an even keel (i.e. zero list and zero trim). If the ship has a list or trim then the ship’s mean draft should be use when entering the curves of form.when entering the curves of form.

H d t ti C ( td )Hydrostatic Curves (cntd..) Keep in mind that all properties on the Hydrostatic Keep in mind that all properties on the Hydrostatic

curves are functions of mean draft and geometry. When weight is added, removed, or shifted, the

operating waterplane and submerged volume change form so that all the geometric properties also change.

0.9

1

MTc

0

0.8

KML

TPc

0.6

0.7

KB

KMt

Dra

ft m

0.4

0.5

LCB

LCF

D

0.2

0.3

Disp.

Wet. Area

WPA

0.10 2000 4000 6000 8000 10000 12000

0 3 6 9 12 15 18 21 24 27

-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

p

Displacement kg

Area m^2

LCB/LCF KB m0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

5 10 15 20 25 30 35 40 45 50 55

0 0.1 0.2

0 0.02 0.04 0.06 0.08 0.1 0.12

KMt m

KML m

Immersion Tonne/cm

Moment to Trim Tonne.m

0.9

1

Waterplane Area

0.7

0.8

Midship Area

Waterplane Area

0.5

0.6

Block

Dra

ft m

0.3

0.4

Prismatic

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 0.9Coeff icients

Tutorial 1Tutorial 1

Tutorial 2

A l f l h 1 0 b 22 h h f ll iA vessel of length 150m, beam 22m has the following waterplane areas at the stated draughts.

Draught (m) 2 4 6 8 10

WPA (m2) 1800 2000 2130 2250 2370

If the lower appendage has a displacement of 2600 pp g ptonnes in water of density 1,025 t/m3 and centre of buoyancy 1.20m above keel, calculate at a draught of 10m the vessel's total displacement KB and C10m the vessel s total displacement, KB and Cb

Other Types of Curvesypi. Sectional Area Curve

The calculated sectional areas (at each stations) also can be represented in curve view.After all the sectional areas are calculated at particular draught, they are plotted in graph.Th h i k S ti l A C h i thThe graph is known as Sectional Area Curve, showing the

curve of sectional areas at each station, particularly at Design draught or design waterline (DWL).g g ( )Sectional Area Curve represents the longitudinal distribution of cross sectional areas at (DWL)

Th di t f ti l l tt d i di t The ordinates of sectional area curve are plotted in distance-squared units

Example: Sectional Area Curve at Waterline 5m

From the curve example, it is clear that the area under the curve represents the volume displacement at

li 5 (DWL)waterline 5m (DWL) Also, displacement and LCB at DWL then can be

determineddetermined

ExerciseExerciseSectional areas of a 180m LBP ship up to 5m draught at constant interval along the length aredraught at constant interval along the length are as follows. Base on the values, create a sectional area curve.

Station 0 1 2 3 4 5 6 7 8 9 10

Area (m2)

5 118 233 291 303 304 304 302 283 171 0

ii. Bonjean Curves The curves of cross sectional area for all stations are

collectively called Bonjean Curves. It showing a set of fair curves formed by plotting of the

areas of transverse sections up to successive waterlinesareas of transverse sections up to successive waterlines At each station along the ships length, a curve of the

transverse shape of the hull is drawn.p The areas of these transverse sections up to each

successive waterline are calculated, and value is plotted on a graphon a graph.

By convention, the Bonjean curves are superimposed onto the ship’s profile.p p

Any predicted waterline required can be drawn on the l t d B j / filcompleted Bonjean curve/profile

One of the principal uses; to determine volume One of the principal uses; to determine volume displacement of ship and its LCB at any draught level, at any trimmed condition

A standard method used is by integrating transverse areas, as learned before.

If the waterline in trim condition the Bonjean Curves are If the waterline in trim condition, the Bonjean Curves are particularly useful.

In the case of trimmed waterline, the trim line maybe ydrawn on the profile of the ship.

Then, drafts are read at which the Bonjean Curve are to be entered.

By drawing a straight line across the contracted profile, the drafts at which the curves are to be read appearthe drafts at which the curves are to be read appear directly at each station.

From there, the values of sectional areas are taken individually at the intersection of the line of drafts drawn and area curves.

All the obtained sectional area values then can be All the obtained sectional area values then can be integrated (eg: Simpson Method) in order to determine the volume of displacement.