Cálculos Geométricos do Casco

Prévia do material em texto

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 p
using this approach. 
Trapezoidal Method
Th 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 Rule
U 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 g
and 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 distance
h is a common interval
Area= )(3
1 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 1st
mmt x h x h
hhd3/1
LCF = hproduct
hhproduct
area
mmtst


3/1
3/1 1
e.g. for 1st
Simpson Rule
hproduct st 
= 
2nd moment, IL = (1/3 x Σproduct 2nd
mmt x h x h2) x 2
area
mmt
product
hproduct 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 at
Aftp p
 For example;
Station ½ ordinate SM Product 
Area
Option1 Option 2 
Lever
Option 3 
Lever
Product 2nd mmt
( )
Amidship
Forward
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 1
For a supertanker, her fully loaded waterplane o a supe a e , e u y oaded a e p a e
has the following ½ ordinates spaced 45m 
apart:p
0, 9.0, 18.1, 23.6, 25.9, 26.2, 22.5, 15.7 and 7.2 
metres respectively.p y
Calculate the waterplane area, WPA and 
waterplane area coefficient, Cwp.p , p
Exercise 2
A water plane of length 270m and breadth 35.5m p g
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 
ti lrespectively.
Calculate;
1.Waterplane area, WPA, and its coefficient, Cwp
2.Longitudinal Centre of Floatation, LCF about the 
id hiamidships.
3.Second moment of area about the amidships
Obtaining Volume
Volumes, 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 t
WL 2
WL 3
various waterlines up to 
required draught, OR
ii) Sectional areas up to the
Waterplane areas at various waterlines
WL 1
ii) Sectional areas up to the 
required draught at various 
stations
Volume 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
3Volume Displacement, (m3)= 1/3 x Σproduct volume x h
Displacement, ∆ (tonne)= Volume Displacement x ρ

1st moment = 1/3 x Σproduct 1st
mmt x h x h
LCB = hproduct st 
1
e.g. for 
1st Simpson 
Rule
LCB = 
volume
mmt
product
hp 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)
Example
A ship length of 150m, breadth 22m has the s p e g o 50 , b ead as e
following waterplane areas at various draught. 
Find the volume, displacement volume and p
vertical 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 y
in 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 midshipsremember that distances aft of amidships are 
negative. 
Immersion or TPCImmersion 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. 
SW
WATPC 
100
 MCTC
To 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
GM
L
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
L
IBM
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
223
1
3
1  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
D
ra
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 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
0.9
1
Waterplane Area
0.7
0.8
Midship Area
Waterplane Area
0.5
0.6
Block
D
ra
ft 
 m
0.3
0.4
Prismatic
0.1
0.2
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Coeff 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 p
tonnes 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 Curvesyp
i. 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 thThe 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
ExerciseExercise
Sectional 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 y
drawn 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.