HYDROSTATICS1 n 2 [Compatibility Mode]

7/27/2019 HYDROSTATICS1 n 2 [Compatibility Mode]

1/41

HYDROSTATICS:

Hull Geometric Calculations


2/41

Fundamental Hull Geometric

a cu a ons

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, , , ,

and VCB

Moreover, all hydrostatic particulars will be calculated

using this approach.


3/41

Trapezoidal Method

e curve s assume to

be represented by a set

.

The area under the curve

is the area of total tra ezoid ABCDEF

Area=


4/41

Simpson Rule

the most popular and common method being used innaval architecture calculations

It is flexible, easy to use, its mathematical basis is easilyunderstood reater accurac and the result reliable.

Its rule states that ship waterlines or sectional area

curves can be represented by polynomials, , ,

moments can be calculated from these polynomials

With Simpson rules, the calculus has been simplified by

us ng mu t p y ng actors or mu t p ers. There are 3 Simpson rules, depending on the number

and location of the offsets.


5/41

1. Simpson 1st Rule

se w en ere s an o

number of offset

three offsets are 1, 4, 1

The multi lier must be in

and end with 1

For more stations (odd

num ers , e mu p ersbecome 1,4,2,4,24,1

This can be roved as

follows:


6/41

where y is a offset distance

h is a common interval


7/41

Area= )(3

1 offsetmultiplierh


8/41

2. Simpson 2nd Rule

Only can be used when number of offsets = 3N+1

s num er o o se

The basic multiplier for set of four offsets are 1, 3, 3, 1

For more stations , the multipliers become

1,3,3,2,3,3,2,3,3,1


9/41

Also the area is preferable to be written as:

8

o setmu t p er


10/41

3. Simpson 3rd Rule

Commonly known as the 5,8-1 rule.

This is to be used when the area between any twoadjacent ordinates is required, three consecutive

.

The multipliers are 5,8,-1.


11/41

Area is the first important geometry that need to be

calculated.

common ypes o area, a erp ane area, an

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

Waterplane area, WPA Sectional Area, AS

, ,comfortable making a tables in solving the calculation


12/41

A B C D E F G H

tat on or nate ro uct

Area

ever ro uct

1st mmt

ever ro uct

2nd mmt

Product

Area

Product

1st mmt

Product

2nd mmt

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

1st moment = 1/3 x product 1st mmt x h x h

LCF =hproduct

pro uct

area

mmtst

3/1

1

e.g. for 1st

Simpson Rulehroduct

=

2nd moment I = 1/3 x roduct nd x h x h2 x 2

area

mmtproduct

1

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


13/41

(datum point). It can be set either zero at aft,

amidship or forward of the ship.If reference point is set at

Aft

For example;Station ordinate SM Product Option1 Option 2 Option 3 Product 2nd mmt

Amidship

Forward

Lever (Product Area x Lever)

AP 1.1 1 1.1 0 -3 6

1 2.7 4 10.8 1 -2 5. . -

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 1

FP 7.7 1 7.7 6 3 0

Product

Area

Product 2nd mmt


14/41

Exercise 1

For a su ertanker her full loaded water lane

has the following ordinates spaced 45ma art:

0, 9.0, 18.1, 23.6, 25.9, 26.2, 22.5, 15.7 and 7.2

metres res ectivel .

Calculate the waterplane area, WPA and

water lane area coefficient Cw .


15/41

Exercise 2

A water plane of length 270m and breadth 35.5m

has 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

respec ve y.

Calculate;1.Waterplane area, WPA, and its coefficient, Cwp

2.Longitudinal Centre of Floatation, LCF about the

am s ps.3.Second moment of area about the amidships


16/41

Obtaining Volume

Volumes, hence

displacement of the ship at

an drau ht can be

calculated if we know either;i) Waterplane areas at WL 2

WL 3

var ous water nes up to

required draught, OR Waterplane areas at various waterlinesWL 1

required draught at various

stations

Volume has its centroid,called longitudinal centre of

buo anc LCB and vertical

centre of buoyancy (VCB)

Sectional areas at various stations


17/41

A B C D E F

Station Station

Area

SM Product

Volume

Lever Product 1st mmt

Product

Volume

Product 1st mmt

Volume Displacement, (m )= 1/3 x product volume x h

Displacement, (tonne)= Volume Displacement x

1st moment = 1/3 x product 1st

mmt x h x h

= hroduct

e.g. for

1st Simpson

Rule

volume

mmtproduct

*h = common interval

(in this case, waterline spacing)


18/41

ec ona areas o a m s p up o m

draught at constant interval along the length are as

.

from amidships.

Area 5 118 233 291 303 304 304 302 283 171 0

m


19/41

Example

A shi len th of 150m breadth 22m has the

following waterplane areas at various draught.Find the volume, dis lacement volume and

vertical centre of buoyancy, VCB at draught

10mDraught (m) 2 4 6 8 10

Waterplanearea, WPA (m2)

1800 2000 2130 2250 2370


20/41

HYDROSTATICS (part II):

H drostatics Particulars and

Curves


21/41

Displacement ()

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 lon itudinal center of buo anc . It is the distancein feet from the longitudinal reference position to the center of

buoyancy. The reference position could be the AP, FP or

.

midships are negative.


22/41

VCB

.

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.

LCF

LCF is the longitudinal center of flotation. It is the distancein from the longitudinal reference to the center of flotation.The reference position could be the AP, FP or amidships. If

s m s ps remem er a s ances a o am s ps arenegative.


23/41

TPC stands for tonnes per centre meter or sometimes justcalled immersion.

TPC is defined as the tonnes required to obtain one centre

meter of parallel sinkage in salt water.ara e s n age s w en e s p c anges s orwar an

after drafts by the same amount so that no change in trim

occurs.

SWWATPC

100


24/41

MCTC

.

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

LMCTC L

100


25/41

This stands for the distance from the keel to thelongitudinal metacenter. For now just assume theme acen er s a conven en re erence po n ver ca y a ovethe keel.

KML= KB + BML

LCFL

IBM

)( midshipmidshipLCF LCFWPAII


26/41

KMT

metacenter. Typically, Naval Architects do not botherputting the subscript T for any property in the transverse

rec on.

=

A B C D E

Station ordinate ( 3 SM Productnd

TI Product

T

2nd mmt

2211 mmtroducthI nd

e.g. is applicable for 1st Simpson Rule


27/41

y ros a c urves All the geometric properties of a ship as a function of

graph for convenience. This graph is called the curves of form or Hydrostatic

.

Each ship has unique curves of form. There are also

tables with the same information which are called the, .

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

even keel (i.e. zero list and zero trim). If the ship has alist or trim then the ships mean draft should be use


28/41

y ros a c urves cn ..

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.


29/41


30/41

0.9

1

MTc

0.8

KML

TPc

0.6

0.7

KB

KMt

raftm

0.4

0.5

LCB

LCF

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

Displacement kg

Area m^2

LCB/LCF KB m

0.4 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


31/41

0.9

1

0.7

0.8

Midship Area

0.5

0.6

Block

Draftm

0.3

0.4

Prismatic

0.1

0.2

0.3 0.4 0.5 0.6 0.7 0.8 0.9Coefficients


32/41


33/41

Tutorial 2

vesse o engt 5 m, eam m as t e o ow ng

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

tonnes in water of density 1,025 t/m3 and centre ofbuoyancy 1.20m above keel, calculate at a draught of

' , b


34/41

Other T es of Curvesi. Sectional Area Curve

The calculated sectional areas (at each stations) also can berepresented in curve view.

After all the sectional areas are calculated at particular

draught, they are plotted in graph.

e grap s nown as ec ona rea urve, s ow ng e

curve of sectional areas at each station, particularly at Design

draught or design waterline (DWL).

Sectional Area Curve represents the longitudinal distributionof cross sectional areas at (DWL)

The ordinates of sectional area curve are plotted in distance-

squared units


35/41

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

water ne m Also, displacement and LCB at DWL then can be


36/41

Sectional areas of a 180m LBP ship up to 5m

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


37/41

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

At each station along the ships length, a curve of the

transverse sha e of the hull is drawn. The areas of these transverse sections up to each

successive waterline are calculated, and value is plotted

on a grap .

By convention, the Bonjean curves are superimposed

onto the shi s rofile.


38/41

Any predicted waterline required can be drawn on the

comp e e on ean curve pro e


39/41

displacement of ship and its LCB at any draught level, atany trimmed condition

A standard method used is by integrating transverse

areas, as learned before.

,

particularly useful.

In the case of trimmed waterline, the trim line maybe

drawn on the profile of the ship.


40/41


41/41

Then, drafts are read at which the Bonjean Curve are to

be entered.

By drawing a straight line across the contracted profile,

directly at each station.

From there, the values of sectional areas are takenindividually at the intersection of the line of drafts drawn

and area curves.

integrated (eg: Simpson Method) in order to determine

the volume of displacement.

Documents

HYDROSTATICS1 n 2 [Compatibility Mode]