The document discusses numerical methods for calculating fundamental hull geometric properties, including the trapezoidal rule and Simpson's rule. It provides details on calculating properties like waterplane area, sectional area, volume, displacement, longitudinal center of buoyancy, and vertical center of buoyancy using these rules. Hydrostatic curves and tables containing geometric properties as a function of draft are also introduced.

1. HYDROSTATICS:
Hull Geometric Calculations

2. Fundamental Hull Geometric
Calculations
 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, LCB
and VCB
 Moreover, all hydrostatic particulars will be calculated
using this approach.

3. Trapezoidal Method
 The curve is assumed to
be represented by a set
of trapezoids.
 The area under the curve
is the area of total trapezoid ABCDEF
Area=

4. 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.
 Its rule states that ship waterlines or sectional area
curves can be represented by polynomials
 Using calculus, the areas, volumes, centroids and
moments can be calculated from these polynomials
 With Simpson rules, the calculus has been simplified by
using multiplying factors or multipliers.
 There are 3 Simpson rules, depending on the number
and location of the offsets.

5. 1. Simpson 1st Rule
 Used when there is an odd
number of offset
 The basic multiplier for set of
three offsets are 1, 4, 1
 The multiplier must begin
and end with 1
 For more stations (odd
numbers), the multipliers
become 1,4,2,4,2……4,1
 This can be proved as
follows:

6. where y is a offset distance
h is a common interval

7. 1  h multiplier offset
Area= 3 ( )

8. 2. Simpson 2nd Rule
 Only can be used when number of offsets = 3N+1
(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
 For more stations , the multipliers become
1,3,3,2,3,3,2……,3,3,1

9.  Also the area is preferable to be written as:
Area = 3 38  h  ( l i li multiplier ff offset
)

10. 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 given.
 The multipliers are 5,8,-1.

11. Obtaining Area
 Area is the first important geometry that need to be
calculated.
 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)
Waterplane area, WPA Sectional Area, AS
 overall for method In overall, applying the Simpson method, it is more
comfortable making a tables in solving the calculation

12. A B C D E F G H
Station ½ ordinate SM Product L P d L P d
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 1
st
mmt x h x h
1/ 3  d product h h
st mmt
 
1/ 3
LCF =  product 
h
area
1
e.g. for 1st
Simpson Rule
product h st
 
=
mmt
2nd moment, IL = ( 1/3 x Σproduct 2
nd
mmt x h x h2) x 2
area
product

1
, L p )
*h = common interval (in this case, station spacing)

13.  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
p p Aft
 For example;
Station ½ ordinate SM Product
Area
Option1 Option 2
Lever
Amidship
Forward
Option 3
Lever
Product 2nd mmt
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 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 1
FP 7.7 1 7.7 6 3 0
ΣProduct
Area
Σ Product 2nd mmt

14. Exercise 1
For a supertanker, o supe a e , her fully loaded waterplane
has the following ½ ordinates spaced 45m
apart:
0, 9.0, 18.1, 23.6, 25.9, 26.2, 22.5, 15.7 and 7.2
metres respectively.
Calculate the waterplane area, WPA and
waterplane area coefficient, Cwp.

15. 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
respectively.
Calculate;
1.Waterplane area, WPA, and its coefficient, Cwp
2.Longitudinal Centre of Floatation, LCF about the
amidships.
3.Second moment of area about the amidships

16. Obtaining Volume
Volumes, hence
displacement of the ship at
any draught can be
calculated if we know either;
i) Waterplane areas at
i t li t
WL 3
WL 2
various waterlines up to
required draught, OR
WL 1
Waterplane areas at various waterlines
ii) Sectional areas up to the
required draught at various
stations
Volume has its centroid,
called longitudinal centre of
buoyancy (LCB) and vertical
centre of buoyancy (VCB)
Sectional areas at various stations

17. A B C D E F
Station Station
Area
SM Product
Volume
Lever Product 1st mmt
ΣProduct
Volume
Σ Product 1st mmt
Volume Displacement,
(m3)= 1/3 x Σproduct volume x h
Displacement, Δ (tonne)= Volume Displacement x ρ
e.g. for
1st Simpson
Rule =
1st moment = 1/3 x Σproduct 1
st
mmt x h x h
LCB  product 1
st
mmt
 h volume
p oduct 
product
*h = common interval
(in this case, waterline spacing)

18. Example
 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 LCB
from amidships.
Station 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)

19. Example
A ship s p length of 150m, breadth 22m has the
following waterplane areas at various draught.
Find the volume, displacement volume and
vertical centre of buoyancy, VCB at draught
10m
Draught (m) 2 4 6 8 10
Waterplane
area, WPA (m2)
1800 2000 2130 2250 2370

20. HYDROSTATICS (part II):
Hydrostatics Particulars and
Curves

21. 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
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 of
midships are negative.

22. VCB
This 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 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 are
negative.

23. 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.
Parallel sinkage is when the ship changes it’’s forward and
after drafts by the same amount so that no change in trim
occurs.
W TPC  A 
SW
100

24.  MCTC
To 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
 MCTC GM
GML
L

100


25.  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 above
the keel.
KML= KB + BML
 LCF
L

BM I
( )2 LCF midship midship I  I WPA LCF

26. KMT
This 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
direction.
KMT= BMT
A B C D E
Station ½ ordinate (½
ordinate) 3
SM Product
KMT = KB + 2nd mmt
 T BM I
2 ΣProduct

T 2nd mmt
I  1   h product nd mmt 
T
p 3 2 2
1
3
e.g. is applicable for 1st Simpson Rule

27. Hydrostatic Curves
 All the geometric properties of a ship as a function of
mean draft have been computed and put into a single
graph for convenience.
 This graph is called the “curves of form” or Hydrostatic
Curves.
 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 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
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.

28. Hydrostatic Curves (cntd..)
 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.

30. 1
0.9
MTc
0.8
0
KML
TPc
0.7
0.6
KB
KMt
Draft m
0.5
0.4
LCB
LCF
D
0.3
0.2
Disp.
WPA
Wet. Area
p
0.10 2000 4000 6000 8000 10000 12000
Displacement kg
0 3 6 9 12 15 18 21 24 27
Area m^2
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
LCB/LCF KB m
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
KMt m
5 10 15 20 25 30 35 40 45 50 55
KML m
0 0.1 0.2
0 0.02 0.04 0.06 0.08 0.1 0.12
Immersion Tonne/cm
Moment to Trim Tonne.m

31. 1
0.9
Waterplane 0.8
0.7
Midship Area
Area
0.6
0.5
Block
Draft m
0.4
0.3
Prismatic
0.2
0.1
0.3 0.4 0.5 0.6 0.7 0.8 0.9
Coefficients

32. Tutorial 1

33. Tutorial 2
A 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
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 Cb

34. Other Types of Curves
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.
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).
Sectional Area Curve represents the longitudinal distribution
of cross sectional areas at (DWL)
 The ordinates of sectional area curve are plotted in distance-squared
units

35. 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
waterline 5m (DWL)
 Also, displacement and LCB at DWL then can be
determined

36. Exercise
Sectional areas of a 180m LBP ship up to 5m
draught 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

37. 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 waterlines
 At each station along the ships length, a curve of the
transverse shape of the hull is drawn.
 The areas of these transverse sections up to each
successive waterline are calculated, and value is plotted
on a graph.
 By convention, the Bonjean curves are superimposed
onto the ship’s profile.

38.  Any predicted waterline required can be drawn on the
completed Bonjean curve/profile

39.  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
particularly useful.
 In the case of trimmed waterline, the trim line maybe
drawn on the profile of the ship.

41.  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 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
integrated (eg: Simpson Method) in order to determine
the volume of displacement.