Simpsons Rules for areas and centroids
85
Rule. In such cases the area or volume should be divided into two parts, the
area of each part being calculated separately, and the total area found by
adding the areas of the two parts together.
Example 1
Show ho
Simpsons Rules for areas and centroids
79
The value of the integral in each case is found by Simpsons Rules using
the areas at equidistant intervals as ordinates; i.e.
h
(A 4 B 2C 4 D E)
3
Volume
or
CI
1
3
Thus the volume of displacement of a ship to an
86
Ship Stability for Masters and Mates
Centroids and centres of gravity
To find the centre of flotation
The centre of flotation is the centre of gravity or centroid of the water-plane
area, and is the point about which a ship heels and trims. It must lie
80
Ship Stability for Masters and Mates
Appendages and intermediate ordinates
Appendages
It has been mentioned previously that areas and volumes calculated by the
use of Simpsons Rules depend for their accuracy on the curvature of
the sides following a de
Simpsons Rules for areas and centroids
77
Example
Find the area of the water-plane described in the first example using Simpsons
Second Rule.
No.
a
b
c
d
e
f
g
1
2
ord.
SM
0
3.7
5.9
7.6
7.5
4.6
0.1
Area function
1
3
3
2
3
3
1
0
11.1
17.7
15.2
22.5
13.8
0.
Simpsons Rules for areas and centroids
In Figure 10.19:
Let KY represent the height of the centre of gravity of volume A above the keel,
and KZ represent the height of the centre of gravity of volume B above the keel.
100
then the area of each water-plane
92
Ship Stability for Masters and Mates
Moments about the keel
KB
Total moment
143 010
3.72 metres
Total volume
38 416
0.531 d
Summary
When using Simpsons Rules for ship calculations always use the following
procedure:
1. Make a sketch using the given
88
Ship Stability for Masters and Mates
Cl
150
25 m
6
A
F
Cl
Cl
Cl
Cl
Cl
Cl
Fig. 10.16
This problem can also be solved by taking the moments about amidships as
in the following example:
Example 2
A ship 75 m long has half-ordinates at the load water-pla
Simpsons Rules for areas and centroids
6 A ship 90 metres long is floating on an even keel at 6 m draft. The halfordinates, commencing from forward, are as follows:
0, 4.88, 6.71, 7.31, 7.01, 6.40 and 0.9 m, respectively.
The half-ordinates 7.5 metres fro
Simpsons Rules for areas and centroids
CI
89
75
9.375 m
8
1 denotes first total.
2 denotes second algebraic total.
The point having a lever of zero is the fulcrum point. All other levers ve
and ve are then relative to this point.
Distance of C.F. from a
78
Ship Stability for Masters and Mates
Example
Three consecutive ordinates in a ships water-plane, spaced 6 metres apart, are
14, 15 and 15.5 m, respectively. Find the area between the last two ordinates.
6m
6m
14 m
15 m
15.5 m
Fig. 10.6(d)
h
(5a 8 b c)
82
Ship Stability for Masters and Mates
Area 1
a
b
h
c
h
Area 2
d
h
e
h
f
h
g
h
h
j
h
Fig. 10.11
erroneous one. To reduce the error the water-plane may be divided into
two parts as shown in the figure.
Then,
Area No. 1 h/3 (a 4 b 2c 4d 2e 4 f g)
To find A
Simpsons Rules for areas and centroids
81
Example
A ships breadths, at 9 m intervals commencing from forward are as follows:
0, 7.6, 8.7 , 9.2, 9.5, 9.4 and 8.5 metres, respectively.
Abaft the last ordinate is an appendage of 50 sq. m. Find the total area
Chapter 11
Second moments of
area moments of
inertia
The second moment of an element of an area about an axis is equal to the
product of the area and the square of its distance from the axis. In some
textbooks, this second moment of area is called the mom
84
Ship Stability for Masters and Mates
1
2
Combined multipliers
1
2
1
2
2
1
4
1
1
1 21
4
2
2
4
4
1
1
2
2
1
2
1 21
2
1
2
Example 2
A ships water-plane is 72 metres long and the lengths of the half-ordinates
commencing from forward are as follows:
0.2, 2
Simpsons Rules for areas and centroids
1
2
ord.
0
3.6
6.0
7.3
7.7
7.6
4.8
2.8
0.6
SM
Area function
1
4
2
4
2
4
1 21
2
0
14.4
12.0
29.2
15.4
30.4
7.2
5.6
0.3
1
2
114.5 1
Area 23 CI 1
CI 100/7 14.29 m
Area of WP 23 14.29 114.5
Ans. Area of WP 1,090.5 sq. m
96
Ship Stability for Masters and Mates
I
dx
b
x
B
A
Fig. 11.3
or
I AB
lb3
3
The theorem of parallel axes
The second moment of an area about an axis through the centroid is equal
to the second moment about any other axis parallel to the first reduced
by
Simpsons Rules for areas and centroids
87
The value of this integral is found by Simpsons Rules using values of the
product x y as ordinates.
Let the distance of the centre of flotation be X from OY, then:
X
Moment
Area
L
2 x y dx
O
L
2 y dx
2
CI
1
O
Exa
Second moments of area moments of inertia
95
Let IAB be the second moment of the whole rectangle about the axis
AB then:
I
dx
x
G
A
B
b
Fig. 11.2
I AB
b/2
b/2
I AB l
lx 2dx
b/2
b/2
x 2dx
x 3 b/2
l
3 b/2
I AB
lb3
12
To find the second moment of a re